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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.
5
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
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).
Sequence in context: A349537 A271919 A201125 * A060409 A305740 A171548
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Jan 17 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 02 2021
STATUS
approved