A154922 - OEIS

%I #9 Mar 02 2021 09:23:10

%S 4,7,7,29,40,29,133,280,280,133,641,2030,2800,2030,641,3157,14630,

%T 28000,28000,14630,3157,15689,102560,278400,360000,278400,102560,

%U 15689,78253,694540,2699900,4557000,4557000,2699900,694540,78253,390881,4549810,25191300,58464000,68040000,58464000,25191300,4549810,390881

%N 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.

%H G. C. Greubel, <a href="/A154922/b154922.txt">Rows n = 0..50 of the triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>

%F 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.

%F 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

%e Triangle begins as:

%e       4;

%e       7,      7;

%e      29,     40,      29;

%e     133,    280,     280,     133;

%e     641,   2030,    2800,    2030,     641;

%e    3157,  14630,   28000,   28000,   14630,    3157;

%e   15689, 102560,  278400,  360000,  278400,  102560,  15689;

%e   78253, 694540, 2699900, 4557000, 4557000, 2699900, 694540, 78253;

%p A154922:= (n,k,p,q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(combinat[stirling2](n, k) + combinat[stirling2](n, n-k));

%p seq(seq(A154922(n,k,2,5), k=0..n), n=0..12); # _G. C. Greubel_, Mar 02 2021

%t T[n_, k_, p_, q_]:= (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS2[n, k] + StirlingS2[n, n-k]);

%t Table[T[n, k, 2, 5], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 2021 *)

%o (Sage)

%o 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))

%o flatten([[A154922(n,k,2,5) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 02 2021

%o (Magma)

%o A154922:= func< n,k,p,q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingSecond(n, k) + StirlingSecond(n, n-k)) >;

%o [A154922(n,k,2,5): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 02 2021

%Y Cf. A154915 (q=1), A154916 (q=3), this sequence (q=5).

%Y Cf. A008277, A048993, A154913, A154914.

%K nonn,tabl,easy,less

%O 0,1

%A _Roger L. Bagula_, Jan 17 2009

%E Edited by _G. C. Greubel_, Mar 02 2021