A358368
a(n) = Sum_{k=0..n} C(n)^2 * binomial(n + k, k), where C(n) is the n-th Catalan number.
Original entry on oeis.org
1, 3, 40, 875, 24696, 814968, 29899584, 1184303835, 49711519000, 2183727606632, 99503164453056, 4672502764108088, 225011739846443200, 11070183993903000000, 554749060302467136000, 28247778810831290434875, 1458696209123375067879000, 76266400563425844598365000
Offset: 0
-
C := n -> binomial(2*n, n)/(n + 1):
A358368 := n -> add(C(n)^2*binomial(n+k,k), k = 0..n): seq(A358368(n), n = 0..17);
# Alternative:
a := proc(n) option remember; if n = 0 then 1 else
(64*n^3 - 32*n^2 - 16*n + 8)*a(n - 1) / (n + 1)^3 fi end: seq(a(n), n = 0..17);
# Third form:
p := n -> hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x):
a := n -> coeff(simplify(p(n)), x, n): seq(a(n), n = 0..17);
-
Array[(2*#+1)*CatalanNumber[#]^3 &, 20, 0] (* Paolo Xausa, Feb 19 2024 *)
A367177
Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).
Original entry on oeis.org
1, 1, 2, 1, 8, 6, 1, 18, 54, 20, 1, 32, 216, 320, 70, 1, 50, 600, 2000, 1750, 252, 1, 72, 1350, 8000, 15750, 9072, 924, 1, 98, 2646, 24500, 85750, 111132, 45276, 3432, 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 2;
[2] 1, 8, 6;
[3] 1, 18, 54, 20;
[4] 1, 32, 216, 320, 70;
[5] 1, 50, 600, 2000, 1750, 252;
[6] 1, 72, 1350, 8000, 15750, 9072, 924;
[7] 1, 98, 2646, 24500, 85750, 111132, 45276, 3432;
[8] 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870;
[9] 1, 162, 7776, 141120, 1111320, 4000752, 6519744, 4447872, 1042470, 48620;
-
p := n -> hypergeom([1/2, -n, -n], [1, 1], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
A367024
Triangle read by rows, T(n, k) = [x^k] -hypergeom([-1/2, -n, -n], [1, 1], 4*x).
Original entry on oeis.org
-1, -1, 2, -1, 8, 2, -1, 18, 18, 4, -1, 32, 72, 64, 10, -1, 50, 200, 400, 250, 28, -1, 72, 450, 1600, 2250, 1008, 84, -1, 98, 882, 4900, 12250, 12348, 4116, 264, -1, 128, 1568, 12544, 49000, 87808, 65856, 16896, 858, -1, 162, 2592, 28224, 158760, 444528, 592704, 342144, 69498, 2860
Offset: 0
Triangle T(n, k) starts:
[0] -1;
[1] -1, 2;
[2] -1, 8, 2;
[3] -1, 18, 18, 4;
[4] -1, 32, 72, 64, 10;
[5] -1, 50, 200, 400, 250, 28;
[6] -1, 72, 450, 1600, 2250, 1008, 84;
[7] -1, 98, 882, 4900, 12250, 12348, 4116, 264;
[8] -1, 128, 1568, 12544, 49000, 87808, 65856, 16896, 858;
[9] -1, 162, 2592, 28224, 158760, 444528, 592704, 342144, 69498, 2860;
-
p := n -> -hypergeom([-1/2, -n, -n], [1, 1], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
A367025
Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = (1 - hypergeom([-1/2, -n - 1, -n - 1], [1, 1], 4*x)) / (2*x).
Original entry on oeis.org
1, 4, 1, 9, 9, 2, 16, 36, 32, 5, 25, 100, 200, 125, 14, 36, 225, 800, 1125, 504, 42, 49, 441, 2450, 6125, 6174, 2058, 132, 64, 784, 6272, 24500, 43904, 32928, 8448, 429, 81, 1296, 14112, 79380, 222264, 296352, 171072, 34749, 1430
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 4, 1;
[2] 9, 9, 2;
[3] 16, 36, 32, 5;
[4] 25, 100, 200, 125, 14;
[5] 36, 225, 800, 1125, 504, 42;
[6] 49, 441, 2450, 6125, 6174, 2058, 132;
[7] 64, 784, 6272, 24500, 43904, 32928, 8448, 429;
[8] 81, 1296, 14112, 79380, 222264, 296352, 171072, 34749, 1430;
[9] 100, 2025, 28800, 220500, 889056, 1852200, 1900800, 868725, 143000, 4862;
-
p := n -> (1 - hypergeom([-1/2, -n-1, -n-1], [1, 1], 4*x)) / (2*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
-
T[n_,k_]:=Binomial[n+1,n-k]^2*Binomial[2*k,k]/(k+1);Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Nov 19 2023 *)
A367178
Triangle read by rows. T(n, k) = binomial(n, k)^2 * CatalanNumber(k).
Original entry on oeis.org
1, 1, 1, 1, 4, 2, 1, 9, 18, 5, 1, 16, 72, 80, 14, 1, 25, 200, 500, 350, 42, 1, 36, 450, 2000, 3150, 1512, 132, 1, 49, 882, 6125, 17150, 18522, 6468, 429, 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430, 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 4, 2;
[3] 1, 9, 18, 5;
[4] 1, 16, 72, 80, 14;
[5] 1, 25, 200, 500, 350, 42;
[6] 1, 36, 450, 2000, 3150, 1512, 132;
[7] 1, 49, 882, 6125, 17150, 18522, 6468, 429;
[8] 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430;
[9] 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862;
-
T := (n, k) -> binomial(n, k)^2 * binomial(2*k, k) / (k + 1):
seq(seq(T(n, k), k = 0..n), n = 0..9);
Showing 1-5 of 5 results.