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%I #13 Mar 02 2021 18:45:03
%S 4,5,5,13,24,13,35,120,120,35,97,546,1008,546,97,275,2310,7200,7200,
%T 2310,275,793,9312,44928,77760,44928,9312,793,2315,36300,255780,
%U 703080,703080,255780,36300,2315,6817,137982,1372356,5660928,8817984,5660928,1372356,137982,6817
%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=3, read by rows.
%H G. C. Greubel, <a href="/A154916/b154916.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=3.
%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=3, 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 5, 5;
%e 13, 24, 13;
%e 35, 120, 120, 35;
%e 97, 546, 1008, 546, 97;
%e 275, 2310, 7200, 7200, 2310, 275;
%e 793, 9312, 44928, 77760, 44928, 9312, 793;
%e 2315, 36300, 255780, 703080, 703080, 255780, 36300, 2315;
%e 6817, 137982, 1372356, 5660928, 8817984, 5660928, 1372356, 137982, 6817;
%p A154916:= (n, k, p, q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(Stirling2(n, k) + Stirling2(n, n-k)):
%p seq(seq(A154916(n,k,2,3), 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, 3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 2021 *)
%o (Sage)
%o 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))
%o flatten([[A154916(n,k,2,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 02 2021
%o (Magma)
%o A154916:= func< n,k,p,q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingSecond(n, k) + StirlingSecond(n, n-k)) >;
%o [A154916(n,k,2,3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 02 2021
%Y Cf. A154915 (q=1), this sequence (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
%E Edited by _G. C. Greubel_, Mar 02 2021
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