A154914 Triangle T(n,m,p,q) = (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS1(n, k) + StirlingS1(n, n-k)) with p=2 and q=3, read by rows.

4, 5, 5, 13, -24, 13, 35, -30, -30, 35, 97, -936, 1584, -936, 97, 275, 2940, -2700, -2700, 2940, 275, 793, -78570, 168012, -194400, 168012, -78570, 793, 2315, 1153350, -2002140, 960120, 960120, -2002140, 1153350, 2315, 6817, -24113544, 46757880, -42378336, 35090496, -42378336, 46757880, -24113544, 6817

Offset

0,1

Links

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

Formula

T(n,m,p,q) = (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS1(n, k) + StirlingS1(n, n-k)) with p=2 and q=3.

Sum_{k=0..n} T(n,k,p,q) = 2*(-p)^n*Pochhammer(-q/p, n) + p^(n+1)*[n < 2], where p=2 and q=3. - G. C. Greubel, Mar 02 2021

Example

Triangle begins as:
     4;
     5,       5;
    13,     -24,       13;
    35,     -30,      -30,      35;
    97,    -936,     1584,    -936,     97;
   275,    2940,    -2700,   -2700,   2940,      275;
   793,  -78570,   168012, -194400, 168012,   -78570,     793;
  2315, 1153350, -2002140,  960120, 960120, -2002140, 1153350, 2315;

Maple

A154914:= (n, k, p, q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(combinat[stirling1](n, k) + combinat[stirling1](n, n-k));
seq(seq(A154914(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))*(StirlingS1[n, k] + StirlingS1[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 A154914(n, k, p, q): return (p^(n-k)*q^k + p^k*q^(n-k))*(stirling_number1(n, k) + stirling_number1(n, n-k))
flatten([[A154914(n, k, 2, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
A154914:= func< n, k, p, q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingFirst(n, k) + StirlingFirst(n, n-k)) >;
[A154914(n, k, 2, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021

Crossrefs

Cf. A154913 (q=1), this sequence (q=3).

Cf. A048994, A154915, A154916, A154922.

Sequence in context: A120132 A331263 A334018 * A154916 A344024 A327703

Adjacent sequences: A154911 A154912 A154913 * A154915 A154916 A154917

Keyword

tabl,sign,easy,less

Author

Roger L. Bagula, Jan 17 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved