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A154916 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=3, read by rows. 5
4, 5, 5, 13, 24, 13, 35, 120, 120, 35, 97, 546, 1008, 546, 97, 275, 2310, 7200, 7200, 2310, 275, 793, 9312, 44928, 77760, 44928, 9312, 793, 2315, 36300, 255780, 703080, 703080, 255780, 36300, 2315, 6817, 137982, 1372356, 5660928, 8817984, 5660928, 1372356, 137982, 6817 (list; table; graph; refs; listen; history; text; internal format)
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=3.
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=3, where T_{n}(x) are the Touchard polynomials (sometimes named Bell polynomials). - G. C. Greubel, Mar 02 2021
EXAMPLE
Triangle begins as:
4;
5, 5;
13, 24, 13;
35, 120, 120, 35;
97, 546, 1008, 546, 97;
275, 2310, 7200, 7200, 2310, 275;
793, 9312, 44928, 77760, 44928, 9312, 793;
2315, 36300, 255780, 703080, 703080, 255780, 36300, 2315;
6817, 137982, 1372356, 5660928, 8817984, 5660928, 1372356, 137982, 6817;
MAPLE
A154916:= (n, k, p, q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(Stirling2(n, k) + Stirling2(n, n-k)):
seq(seq(A154916(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))*(StirlingS2[n, k] + StirlingS2[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 A154916(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([[A154916(n, k, 2, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
A154916:= func< n, k, p, q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingSecond(n, k) + StirlingSecond(n, n-k)) >;
[A154916(n, k, 2, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021
CROSSREFS
Cf. A154915 (q=1), this sequence (q=3), A154922 (q=5).
Sequence in context: A331263 A334018 A154914 * A344024 A327703 A077061
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Jan 17 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 02 2021
STATUS
approved

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Last modified April 14 07:58 EDT 2024. Contains 371655 sequences. (Running on oeis4.)