Catalanpaths
A053121 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ is(k = 0)\ ? \ 0 : \frac{k+1}{n+1} \binom{n+1}{(n-k)/2} } \)
A000007 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (1 - [2 | k])\) ➤ RevOddSum ➤ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A000012 \(T_{n , n}\) ➤ 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 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---------- \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A000079 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 4 8 16 32 64 128 256 512 1024 2048
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 4 8 16 32 64 128 256 512 1024 2048
A000245 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 1 0 3 0 9 0 28 0 90 0 297 0 1001 0 3432
A001405 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 2 3 6 10 20 35 70 126 252 462 924
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 2 -3 6 -10 20 -35 70 -126 252 -462
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 2 3 6 10 20 35 70 126 252 462 924
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 2 3 6 10 20 35 70 126 252 462 924
A002057 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 1 0 4 0 14 0 48 0 165 0 572 0 2002 0
A002522 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 1 2 5 10 17 26 37 50 65 82 101 122 145
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 1 2 5 10 17 26 37 50 65 82 101 122 145
A026005 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 0 4 0 27 0 208 0 1700 0 14364 0
A045621 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 2 5 10 22 44 93 186 386 772 1586
A052173 \(T_{n, n - k}\) ➤ Trev ➤ 1 1 0 1 0 1 1 0 2 0 1 0 3 0 2 1 0 4 0 5
---------- \(T_{n, n-k}\ (-1)^{n-k}\) ➤ RevTalt ➤ 1 1 0 1 0 1 1 0 2 0 1 0 3 0 2 1 0 4 0 5
A053121 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 1 0 1 0 2 0 1 2 0 3 0 1 0 5 0 4 0
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 1 0 1 0 -2 0 -1 2 0 3 0 1 0 -5 0
A054341 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 5 12 30 74 185 460 1150 2868 7170
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 5 -12 30 -74 185 -460 1150 -2868
A054602 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 3 12 33 72 135 228 357 528 747 1020
A058331 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 1 3 9 19 33 51 73 99 129 163 201 243
A101461 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 1 2 3 5 9 14 28 48 90 165 297 572
A107430 \(\sum_{j=0}^{n-k} T_{n, n-j}\) ➤ RevTacc ➤ 1 1 1 1 1 2 1 1 3 3 1 1 4 4 6 1 1 5 5
A112552 \(T^{-1}_{n + 1, k + 1}\) ➤ Tinv11 ➤ 1 0 1 -2 0 1 0 -3 0 1 3 0 -4 0 1 0 6 0
A112554 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 0 1 2 0 1 0 3 0 1 5 0 4 0 1 0 9 0 5 0
A121724 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 5 9 45 97 485 1145 5725 14289 71445
---------- \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 5 9 45 97 485 1145 5725 14289 71445
A121725 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 10 19 190 442 4420 11395 113950
A126120 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 0 1 0 2 0 5 0 14 0 42 0 132 0 429 0
---------- \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 0 2 0 5 0 14 0 42 0 132 0 429 0 1430
---------- \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 2 0 5 0 14 0 42 0 132 0 429 0 1430
A126596 \(T_{2 n, n}\) ➤ CentralE ➤ 1 0 3 0 20 0 154 0 1260 0 10659 0 92092
A126869 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 2 0 6 0 20 0 70 0 252 0 924 0 3432
A126931 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 10 33 110 366 1220 4065 13550 45162
A138364 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 0 3 0 10 0 35 0 126 0 462 0 1716 0
A162515 \(T^{-1}_{n, n - k}\) ➤ Trevinv ➤ 1 1 0 1 0 -1 1 0 -2 0 1 0 -3 0 1 1 0 -4
A168561 \(T^{-1}_{n, k}\) ➤ Tinv ➤ 1 0 1 -1 0 1 0 -2 0 1 1 0 -3 0 1 0 3 0
A274112 \(\sum_{k=0}^{n/2} T_{n - k, n-k}\) ➤ RevAntiDSum ➤ 1 1 1 1 2 3 4 5 8 12 17 23 35 52 75 105
A296663 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 4 7 20 38 96 187 444 874 2000 3958
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 4 7 20 38 96 187 444 874 2000 3958
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 4 7 20 38 96 187 444 874 2000 3958
A297382 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1
A344500 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 2 7 21 66 216 715 2395 8101 27598
---------- \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 2 7 21 66 216 715 2395 8101 27598
A359108 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 2 0 14 0 110 0 910 0 7752 0 67298 0