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A176666 A triangle of polynomial coefficients:p(x,n)=Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}].

%I A176666 #4 Jul 22 2025 07:59:06
%S A176666 1,1,3,1,-16,25,1,588,-904,343,1,-35108,65593,-36965,6561,1,3541662,
%T A176666 -7450307,5299298,-1551461,161051,1,-539667860,1239476145,-1027098387,
%U A176666 393094596,-70630574,4826809,1,115929493398,-285126982237,264011385389
%N A176666 A triangle of polynomial coefficients:p(x,n)=Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}].
%C A176666 Row sums are:A103457;
%C A176666 {1, 4, 10, 28, 82, 244, 730, 2188, 6562, 19684, 59050,...}.
%F A176666 p(x,n)=Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}];
%F A176666 t(n,m)=coefficients(p(x,n))
%e A176666 {1},
%e A176666 {1, 3},
%e A176666 {1, -16, 25},
%e A176666 {1, 588, -904, 343},
%e A176666 {1, -35108, 65593, -36965, 6561},
%e A176666 {1, 3541662, -7450307, 5299298, -1551461, 161051},
%e A176666 { 1, -539667860, 1239476145, -1027098387, 393094596, -70630574, 4826809},
%e A176666 {1, 115929493398, -285126982237, 264011385389, -120438105421, 28978650041, -3525298358, 170859375},
%e A176666 {1, -33405526460804, 86851508060145, -87619801707127, 45402414077950, -13236000326919, 2193188923598, -192758317723, 6975757441},
%e A176666 {1, 12439546100725062, -33876724511327305, 36619991865553925, -20936375400104384, 7018154767854372, -1426322806941012, 172905465804793, -11498169243547, 322687697779},
%e A176666 {1, -5815351979718349460, 16476663041157314889, -18861838035155184791, 11671607490973992658, -4358525114199083475, 1028212770824559839, -154333184246062051, 14292794059654483, -744463577761244, 16679880978201}
%t A176666 Clear[p, x, n]
%t A176666 p[x_, n_] := Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}];
%t A176666 Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
%t A176666 Flatten[%]
%Y A176666 Cf. A103457
%K A176666 sign,tabl,uned
%O A176666 0,3
%A A176666 _Roger L. Bagula_, Apr 23 2010