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%I #5 Mar 15 2021 21:30:45
%S 3,5,5,9,27,9,17,102,102,17,33,330,660,330,33,65,975,3250,3250,975,65,
%T 129,2709,13545,22575,13545,2709,129,257,7196,50372,125930,125930,
%U 50372,7196,257,513,18468,172368,603288,904932,603288,172368,18468,513
%N Triangle T(n,k,p,q) = (p^n + q^n)*A001263(n, k) with p=2 and q=1, read by rows.
%H G. C. Greubel, <a href="/A155537/b155537.txt">Rows n = 1..50 of the triangle, flattened</a>
%F 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.
%F From _G. C. Greubel_, Mar 15 2021: (Start)
%F T(n,k,p,q) = T(n,k,q,p) = (p^n + q^n)*A001263(n, k).
%F T(n,k,2,1) = A000051(n) * A001263(n,k).
%F Sum_{k=1..n} T(n,k,p,q) = (p^n + q^n)*C(n), where C(n) are the Catalan numbers (A000108). (End)
%e Triangle begins as:
%e 3;
%e 5, 5;
%e 9, 27, 9;
%e 17, 102, 102, 17;
%e 33, 330, 660, 330, 33;
%e 65, 975, 3250, 3250, 975, 65;
%e 129, 2709, 13545, 22575, 13545, 2709, 129;
%e 257, 7196, 50372, 125930, 125930, 50372, 7196, 257;
%e 513, 18468, 172368, 603288, 904932, 603288, 172368, 18468, 513;
%e 1025, 46125, 553500, 2583000, 5424300, 5424300, 2583000, 553500, 46125, 1025;
%p A155537:= (n,k,p,q)-> (p^n + q^n)*binomial(n-1, k-1)*binomial(n, k)/(n-k+1);
%p seq(seq(A155537(n,k,2,1), k=1..n), n=1..12); # _G. C. Greubel_, Mar 15 2021
%t 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);
%t Table[T[n,k,2,1], {n, 12}, {k, n}]//Flatten (* modified by _G. C. Greubel_, Mar 15 2021 *)
%o (Sage)
%o def T(n,k,p,q): return (p^n + q^n)*binomial(n-1, k-1)*binomial(n, k)/(n-k+1)
%o flatten([[T(n,k,2,1) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 15 2021
%o (Magma)
%o T:= func< n,k,p,q | (p^n + q^n)*Binomial(n-1, k-1)*Binomial(n, k)/(n-k+1) >;
%o [T(n,k,2,1): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 15 2021
%Y Cf. A000051, A000108, A001263.
%K nonn,tabl
%O 1,1
%A _Roger L. Bagula_, Jan 23 2009
%E Edited by _G. C. Greubel_, Mar 15 2021
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