A154922 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=5, read by rows.

4, 7, 7, 29, 40, 29, 133, 280, 280, 133, 641, 2030, 2800, 2030, 641, 3157, 14630, 28000, 28000, 14630, 3157, 15689, 102560, 278400, 360000, 278400, 102560, 15689, 78253, 694540, 2699900, 4557000, 4557000, 2699900, 694540, 78253, 390881, 4549810, 25191300, 58464000, 68040000, 58464000, 25191300, 4549810, 390881

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=5.

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=5, where T_{n}(x) are the Touchard polynomials (sometimes named Bell polynomials). - G. C. Greubel, Mar 02 2021

Example

Triangle begins as:
      4;
      7,      7;
     29,     40,      29;
    133,    280,     280,     133;
    641,   2030,    2800,    2030,     641;
   3157,  14630,   28000,   28000,   14630,    3157;
  15689, 102560,  278400,  360000,  278400,  102560,  15689;
  78253, 694540, 2699900, 4557000, 4557000, 2699900, 694540, 78253;

Maple

A154922:= (n, k, p, q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(combinat[stirling2](n, k) + combinat[stirling2](n, n-k));
seq(seq(A154922(n, k, 2, 5), 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, 5], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)

Prog

(Sage)
def A154922(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([[A154922(n, k, 2, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
A154922:= func< n, k, p, q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingSecond(n, k) + StirlingSecond(n, n-k)) >;
[A154922(n, k, 2, 5): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021

Crossrefs

Cf. A154915 (q=1), A154916 (q=3), this sequence (q=5).

Cf. A008277, A048993, A154913, A154914.

Sequence in context: A349537 A271919 A201125 * A060409 A305740 A171548

Adjacent sequences: A154919 A154920 A154921 * A154923 A154924 A154925

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Jan 17 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved