A154916 Triangle T(n,m,p,q) = (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS2(n, k) + StirlingS2(n, n-k)) with p=2 and q=3, read by rows.

4, 5, 5, 13, 24, 13, 35, 120, 120, 35, 97, 546, 1008, 546, 97, 275, 2310, 7200, 7200, 2310, 275, 793, 9312, 44928, 77760, 44928, 9312, 793, 2315, 36300, 255780, 703080, 703080, 255780, 36300, 2315, 6817, 137982, 1372356, 5660928, 8817984, 5660928, 1372356, 137982, 6817

Offset

0,1

Links

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

Eric Weisstein's World of Mathematics, Bell Polynomial

Formula

T(n,m,p,q) = (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS2(n, k) + StirlingS2(n, n-k)) with p=2 and q=3.

Sum_{k=0..n} T(n,k,p,q) = 2*p^n*( T_{n}(q/p) + (q/p)^n*T_{n}(p/q) ), with p=2 and q=3, where T_{n}(x) are the Touchard polynomials (sometimes named Bell polynomials). - G. C. Greubel, Mar 02 2021

Example

Triangle begins as:
     4;
     5,      5;
    13,     24,      13;
    35,    120,     120,      35;
    97,    546,    1008,     546,      97;
   275,   2310,    7200,    7200,    2310,     275;
   793,   9312,   44928,   77760,   44928,    9312,     793;
  2315,  36300,  255780,  703080,  703080,  255780,   36300,   2315;
  6817, 137982, 1372356, 5660928, 8817984, 5660928, 1372356, 137982, 6817;

Maple

A154916:= (n, k, p, q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(Stirling2(n, k) + Stirling2(n, n-k)):
seq(seq(A154916(n, k, 2, 3), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021

Mathematica

T[n_, k_, p_, q_]:= (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS2[n, k] + StirlingS2[n, n-k]);
Table[T[n, k, 2, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)

Prog

(Sage)
def A154916(n, k, p, q): return (p^(n-k)*q^k + p^k*q^(n-k))*(stirling_number2(n, k) + stirling_number2(n, n-k))
flatten([[A154916(n, k, 2, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
A154916:= func< n, k, p, q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingSecond(n, k) + StirlingSecond(n, n-k)) >;
[A154916(n, k, 2, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021

Crossrefs

Cf. A154915 (q=1), this sequence (q=3), A154922 (q=5).

Cf. A008277, A048993, A154913, A154914.

Sequence in context: A331263 A334018 A154914 * A344024 A327703 A077061

Adjacent sequences: A154913 A154914 A154915 * A154917 A154918 A154919

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Jan 17 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved