login
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
OFFSET
1,1
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Jan 23 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 15 2021
STATUS
approved