A154913 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=1, read by rows.

4, 3, 3, 5, -8, 5, 9, -6, -6, 9, 17, -120, 176, -120, 17, 33, 252, -180, -180, 252, 33, 65, -4590, 7180, -7200, 7180, -4590, 65, 129, 46134, -57204, 21336, 21336, -57204, 46134, 129, 257, -658840, 910520, -603680, 433216, -603680, 910520, -658840, 257

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=1.

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=1. - G. C. Greubel, Mar 02 2021

Example

Triangle begins as:
    4;
    3,       3;
    5,      -8,      5;
    9,      -6,     -6,       9;
   17,    -120,    176,    -120,     17;
   33,     252,   -180,    -180,    252,      33;
   65,   -4590,   7180,   -7200,   7180,   -4590,     65;
  129,   46134, -57204,   21336,  21336,  -57204,  46134,     129;
  257, -658840, 910520, -603680, 433216, -603680, 910520, -658840, 257;

Maple

A154913:= (n, k, p, q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(Stirling1(n, k) + Stirling1(n, n-k));
seq(seq(A154913(n, k, 2, 1), 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, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)

Prog

(Sage)
def A154913(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([[A154913(n, k, 2, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
A154913:= func< n, k, p, q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingFirst(n, k) + StirlingFirst(n, n-k)) >;
[A154913(n, k, 2, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021

Crossrefs

Cf. this sequence (q=1), A154914 (q=3).

Cf. A048994, A154915, A154916, A154922.

Sequence in context: A362123 A052360 A263046 * A154915 A238376 A237197

Adjacent sequences: A154910 A154911 A154912 * A154914 A154915 A154916

Keyword

tabl,sign,easy,less

Author

Roger L. Bagula, Jan 17 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved