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%I #16 Mar 02 2021 18:44:34
%S 4,3,3,5,-8,5,9,-6,-6,9,17,-120,176,-120,17,33,252,-180,-180,252,33,
%T 65,-4590,7180,-7200,7180,-4590,65,129,46134,-57204,21336,21336,
%U -57204,46134,129,257,-658840,910520,-603680,433216,-603680,910520,-658840,257
%N Triangle T(n,m,p,q) = (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS1(n, k) + StirlingS1(n, n-k)) with p=2 and q=1, read by rows.
%H G. C. Greubel, <a href="/A154913/b154913.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n,m,p,q) = (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingS1(n, k) + StirlingS1(n, n-k)) with p=2 and q=1.
%F Sum_{k=0..n} T(n,k,p,q) = 2*(-p)^n*Pochhammer(-q/p, n) + p^(n+1)*[n < 2], where p=2 and q=1. - _G. C. Greubel_, Mar 02 2021
%e Triangle begins as:
%e 4;
%e 3, 3;
%e 5, -8, 5;
%e 9, -6, -6, 9;
%e 17, -120, 176, -120, 17;
%e 33, 252, -180, -180, 252, 33;
%e 65, -4590, 7180, -7200, 7180, -4590, 65;
%e 129, 46134, -57204, 21336, 21336, -57204, 46134, 129;
%e 257, -658840, 910520, -603680, 433216, -603680, 910520, -658840, 257;
%p A154913:= (n, k, p, q) -> (p^(n-k)*q^k + p^k*q^(n-k))*(Stirling1(n, k) + Stirling1(n, n-k));
%p seq(seq(A154913(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))*(StirlingS1[n, k] + StirlingS1[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 A154913(n,k,p,q): return (p^(n-k)*q^k + p^k*q^(n-k))*(stirling_number1(n, k) + stirling_number1(n, n-k))
%o flatten([[A154913(n,k,2,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 02 2021
%o (Magma)
%o A154913:= func< n,k,p,q | (p^(n-k)*q^k + p^k*q^(n-k))*(StirlingFirst(n, k) + StirlingFirst(n, n-k)) >;
%o [A154913(n,k,2,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 02 2021
%Y Cf. this sequence (q=1), A154914 (q=3).
%Y Cf. A048994, A154915, A154916, A154922.
%K tabl,sign,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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