A154915 - OEIS

%I #15 Mar 02 2021 09:22:55

%S 4,3,3,5,8,5,9,24,24,9,17,70,112,70,17,33,198,480,480,198,33,65,544,

%T 1920,2880,1920,544,65,129,1452,7308,15624,15624,7308,1452,129,257,

%U 3770,26724,80640,108864,80640,26724,3770,257,513,9546,94644,408312,706608,706608,408312,94644,9546,513

%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=1, read by rows.

%H G. C. Greubel, <a href="/A154915/b154915.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=1.

%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=1, 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     3,    3;

%e     5,    8,     5;

%e     9,   24,    24,      9;

%e    17,   70,   112,     70,     17;

%e    33,  198,   480,    480,    198,     33;

%e    65,  544,  1920,   2880,   1920,    544,     65;

%e   129, 1452,  7308,  15624,  15624,   7308,   1452,   129;

%e   257, 3770, 26724,  80640, 108864,  80640,  26724,  3770,  257;

%e   513, 9546, 94644, 408312, 706608, 706608, 408312, 94644, 9546, 513;

%p A154915:= (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(A154915(n,k,2,1), 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, 1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 2021 *)

%o (Sage)

%o 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))

%o flatten([[A154915(n,k,2,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 02 2021

%o (Magma)

%o A154915:= func< n,k,p,q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingSecond(n, k) + StirlingSecond(n, n-k)) >;

%o [A154915(n,k,2,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 02 2021

%Y Cf. this sequence (q=1), A154916 (q=3), A154922 (q=5).

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

%K nonn,tabl,easy,less

%O 0,1

%A _Roger L. Bagula_, Jan 17 2009