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A155537 Triangle T(n,k,p,q) = (p^n + q^n)*A001263(n, k) with p=2 and q=1, read by rows. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified November 27 13:09 EST 2022. Contains 358405 sequences. (Running on oeis4.)