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A156901 Triangle formed by coefficients of the expansion of p(x, n), where p(x,n) = (1 + 2*x - x^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-2*x + x^2)^j.

%I A156901 #7 Jan 07 2022 13:05:25
%S A156901 1,1,1,-2,1,1,-8,8,-4,1,1,-22,55,-52,23,-6,1,1,-52,290,-472,394,-188,
%T A156901 50,-8,1,1,-114,1265,-3624,4838,-3668,1750,-536,97,-10,1,1,-240,4884,
%U A156901 -24092,49239,-56448,40664,-19320,6231,-1360,180,-12,1,1,-494,17419,-142124,441625,-730898,749723,-515944,247067,-83122,19673,-3244,331,-14,1
%N A156901 Triangle formed by coefficients of the expansion of p(x, n), where p(x,n) = (1 + 2*x - x^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-2*x + x^2)^j.
%H A156901 G. C. Greubel, <a href="/A156901/b156901.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A156901 T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1 + 2*x - x^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-2*x + x^2)^j.
%F A156901 T(n, 1) = (-1)*A005803(n) for n >= 2.
%e A156901 Irregular triangle begins as:
%e A156901   1;
%e A156901   1;
%e A156901   1,   -2,    1;
%e A156901   1,   -8,    8,     -4,     1;
%e A156901   1,  -22,   55,    -52,    23,     -6,     1;
%e A156901   1,  -52,  290,   -472,   394,   -188,    50,     -8,    1;
%e A156901   1, -114, 1265,  -3624,  4838,  -3668,  1750,   -536,   97,   -10,   1;
%e A156901   1, -240, 4884, -24092, 49239, -56448, 40664, -19320, 6231, -1360, 180, -12, 1;
%t A156901 p[x_, n_]= (1+2*x-x^2)^(n+1)*Sum[(k+1)^n*(-2*x+x^2)^k, {k,0,Infinity}];
%t A156901 Table[CoefficientList[p[x, n], x], {n,0,10}]//Flatten
%o A156901 (Sage)
%o A156901 def T(n, k): return ( (1+2*x-x^2)^(n+1)*sum((j+1)^n*(x^2-2*x)^j for j in (0..2*n+1)) ).series(x, 2*n+2).list()[k]
%o A156901 flatten([1]+[[T(n, k) for k in (0..2*n-2)] for n in (1..12)]) # _G. C. Greubel_, Jan 07 2022
%Y A156901 Cf. A005803, A156890, A156896, A156918.
%K A156901 sign,tabf
%O A156901 0,4
%A A156901 _Roger L. Bagula_, Feb 17 2009
%E A156901 Edited by _G. C. Greubel_, Jan 07 2022