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%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
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