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=1.
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=1, where T_{n}(x) are the Touchard polynomials (sometimes named Bell polynomials). - G. C. Greubel, Mar 02 2021
EXAMPLE
Triangle begins as:
4;
3, 3;
5, 8, 5;
9, 24, 24, 9;
17, 70, 112, 70, 17;
33, 198, 480, 480, 198, 33;
65, 544, 1920, 2880, 1920, 544, 65;
129, 1452, 7308, 15624, 15624, 7308, 1452, 129;
257, 3770, 26724, 80640, 108864, 80640, 26724, 3770, 257;
513, 9546, 94644, 408312, 706608, 706608, 408312, 94644, 9546, 513;
MAPLE
A154915:= (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(A154915(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))*(StirlingS2[n, k] + StirlingS2[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 A154915(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([[A154915(n, k, 2, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
A154915:= func< n, k, p, q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingSecond(n, k) + StirlingSecond(n, n-k)) >;
[A154915(n, k, 2, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Jan 17 2009
STATUS
approved