A155537 Triangle T(n,k,p,q) = (p^n + q^n)*A001263(n, k) with p=2 and q=1, read by rows.

3, 5, 5, 9, 27, 9, 17, 102, 102, 17, 33, 330, 660, 330, 33, 65, 975, 3250, 3250, 975, 65, 129, 2709, 13545, 22575, 13545, 2709, 129, 257, 7196, 50372, 125930, 125930, 50372, 7196, 257, 513, 18468, 172368, 603288, 904932, 603288, 172368, 18468, 513

Offset

1,1

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

Define T(n,k,p,q) = (p^n + q^n)*binomial(n-1, k-1)*binomial(n, k)/(n-k+1) (A scaled Narayana triangle) for various p and q. When p = 2 and q = 1 this sequence is obtained.

From G. C. Greubel, Mar 15 2021: (Start)

T(n,k,p,q) = T(n,k,q,p) = (p^n + q^n)*A001263(n, k).

T(n,k,2,1) = A000051(n) * A001263(n,k).

Sum_{k=1..n} T(n,k,p,q) = (p^n + q^n)*C(n), where C(n) are the Catalan numbers (A000108). (End)

Example

Triangle begins as:
     3;
     5,     5;
     9,    27,      9;
    17,   102,    102,      17;
    33,   330,    660,     330,      33;
    65,   975,   3250,    3250,     975,      65;
   129,  2709,  13545,   22575,   13545,    2709,     129;
   257,  7196,  50372,  125930,  125930,   50372,    7196,    257;
   513, 18468, 172368,  603288,  904932,  603288,  172368,  18468,   513;
  1025, 46125, 553500, 2583000, 5424300, 5424300, 2583000, 553500, 46125, 1025;

Maple

A155537:= (n, k, p, q)-> (p^n + q^n)*binomial(n-1, k-1)*binomial(n, k)/(n-k+1);
seq(seq(A155537(n, k, 2, 1), k=1..n), n=1..12); # G. C. Greubel, Mar 15 2021

Mathematica

T[n_, k_, p_, q_]:= T[n, k, p, q]= (p^n + q^n)*Binomial[n-1, k-1]*Binomial[n, k]/(n-k+1);
Table[T[n, k, 2, 1], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 15 2021 *)

Prog

(Sage)
def T(n, k, p, q): return (p^n + q^n)*binomial(n-1, k-1)*binomial(n, k)/(n-k+1)
flatten([[T(n, k, 2, 1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 15 2021
(Magma)
T:= func< n, k, p, q | (p^n + q^n)*Binomial(n-1, k-1)*Binomial(n, k)/(n-k+1) >;
[T(n, k, 2, 1): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 15 2021

Crossrefs

Cf. A000051, A000108, A001263.

Sequence in context: A145282 A049757 A190864 * A164663 A098971 A093572

Adjacent sequences: A155534 A155535 A155536 * A155538 A155539 A155540

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 23 2009

Extensions

Edited by G. C. Greubel, Mar 15 2021

Status

approved