Narayana
A090181 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ \binom{n}{n-k} \binom{n-1}{n-k} \frac{1}{n-k+1} } \)
A000007 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A000012 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000027 \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A000108 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 2 5 14 42 132 429 1430 4862 16796
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 5 14 42 132 429 1430 4862 16796
A000217 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 3 6 10 15 21 28 36 45 55 66 78 91 105
---------- \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 3 6 10 15 21 28 36 45 55 66 78 91
A000891 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 3 20 175 1764 19404 226512 2760615
A001003 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 3 11 45 197 903 4279 20793 103049
A001263 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 1 1 3 1 1 6 6 1 1 10 20 10 1 1 15
---------- \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 1 1 1 3 1 1 6 6 1 1 10 20 10 1 1 15
A001700 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 3 10 35 126 462 1716 6435 24310
---------- \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 3 10 35 126 462 1716 6435 24310
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 3 10 35 126 462 1716 6435 24310
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 3 10 35 126 462 1716 6435 24310
A002054 \(\sum_{k=0}^{n} T_{n, n-k}\ k\) ➤ RevTransNat0 ➤ 0 0 1 5 21 84 330 1287 5005 19448 75582
A002378 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 2 6 12 20 30 42 56 72 90 110 132 156
A002415 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 6 20 50 105 196 336 540 825 1210 1716
---------- \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 1 6 20 50 105 196 336 540 825 1210
A004148 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 2 4 8 17 37 82 185 423 978 2283
---------- \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 2 4 8 17 37 82 185 423 978 2283
A005558 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 3 6 20 50 175 490 1764 5292 19404
---------- \(T_{n, n / 2}\) ➤ RevColMiddle ➤ 1 1 1 3 6 20 50 175 490 1764 5292 19404
A006318 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 6 22 90 394 1806 8558 41586 206098
A006542 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 1 10 50 175 490 1176 2520 4950 9075
A007564 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 4 19 100 562 3304 20071 124996
A028387 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 5 11 19 29 41 55 71 89 109 131 155
A033445 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 5 22 57 116 205 330 497 712 981 1310
A034267 \(\sum_{k=0}^{n} T_{n, n-k}\ k^{2}\) ➤ RevTransSqrs ➤ 0 0 1 7 39 196 930 4257 19019 83512
A046715 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 1 10 105 1176 13860 169884 2147145
A047891 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 12 57 300 1686 9912 60213 374988
A071684 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 2 7 22 66 212 715 2438 8398 29372
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 1 2 7 22 66 212 715 2438 8398 29372
A071688 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 1 3 7 20 66 217 715 2424 8398 29414
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 1 3 7 20 66 217 715 2424 8398 29414
A090181 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 1 0 1 3 1 0 1 6 6 1 0 1 10 20
---------- \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 1 0 1 3 1 0 1 6 6 1 0 1 10 20
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 1 0 -1 3 -1 0 -1 6 -6 1 0
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 -1 0 1 -3 1 0 1 -6 6 -1 0 1 -10
A091593 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 -1 -1 5 -3 -21 51 41 -391 407 1927
A103364 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 -1 1 2 -3 1 -7 12 -6 1 39 -70 40 -10
---------- \((T_{n + 1, n - k + 1})^{-1}\) ➤ Tinvrev11 ➤ 1 -1 1 2 -3 1 -7 12 -6 1 39 -70 40 -10
A125558 \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 6 50 490 5292 60984 736164 9202050
A126120 \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 0 1 0 -2 0 5 0 -14 0 42 0 -132 0
A132813 \(T_{n + 1, k + 1}\ (k + 1) \) ➤ Tder ➤ 1 1 2 1 6 3 1 12 18 4 1 20 60 40 5 1 30
A141222 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 5 22 95 406 1722 7260 30459 127270
A152681 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 2 2 -10 6 42 -102 -82 782 -814
A189176 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 5 15 49 168 594 2145 7865 29172
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 5 15 49 168 594 2145 7865 29172
A242369 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 3 19 185 2426 39907 788019 18130401
A349740 \(\sum_{j=0}^{k} T_{n, j}\) ➤ Tacc ➤ 1 0 1 0 1 2 0 1 4 5 0 1 7 13 14 0 1 11