A157118 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1, read by rows.

2, 1, 1, 1, 6, 1, 1, 27, 27, 1, 1, 88, 672, 88, 1, 1, 225, 9150, 9150, 225, 1, 1, 486, 98385, 395352, 98385, 486, 1, 1, 931, 1126951, 11748681, 11748681, 1126951, 931, 1, 1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1, 1, 2673, 201755880, 16093941435, 32251030119, 32251030119, 16093941435, 201755880, 2673, 1

Offset

0,1

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1.

T(n, n-k) = T(n, k).

Example

Triangle begins as:
  2;
  1,    1;
  1,    6,        1;
  1,   27,       27,         1;
  1,   88,      672,        88,         1;
  1,  225,     9150,      9150,       225,         1;
  1,  486,    98385,    395352,     98385,       486,        1;
  1,  931,  1126951,  11748681,  11748681,   1126951,      931,    1;
  1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1;

Mathematica

A001263[n_, k_]:= Binomial[n-1, k-1]*Binomial[n, k]/(n-k+1);
f[n_, k_]:= If[k<=n, A001263[n*k+1, n-k+1], A001263[n*(n-k)+1, k+1]];
T[n_, k_]:= If[n==1, 1, f[n, k] + f[n, n-k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)

Prog

(Magma)
A001263:= func< n, k | Binomial(n-1, k-1)*Binomial(n, k)/(n-k+1) >;
f:= func< n, k | k le n select A001263(n*k+1, n-k+1) else A001263(n*(n-k)+1, k+1) >;
A157118:= func< n, k | n eq 1 select 1 else f(n, k) + f(n, n-k) >;
[A157118(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022
(Sage)
def A001263(n, k): return binomial(n-1, k-1)*binomial(n, k)/(n-k+1)
def f(n, k): return A001263(n*k+1, n-k+1) if (k<n+1) else A001263(n*(n-k)+1, k+1)
def A157118(n, k): return 1 if (n==1) else f(n, k) + f(n, n-k)
flatten([[A157118(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022

Crossrefs

Cf. A001263, A157114, A157117.

Sequence in context: A369964 A326047 A280491 * A156186 A156233 A251725

Adjacent sequences: A157115 A157116 A157117 * A157119 A157120 A157121

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 23 2009

Extensions

Edited by G. C. Greubel, Jan 11 2022

Status

approved