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A156817 A triangle sequence from polynomial coefficients: p(x, n) = (4*(n - 1) + (4(n - 1))*x^2)*p(x, n - 1) + 2*(n - 1)x^2*p(x, n - 2). 0

%I #2 Oct 12 2012 14:54:56

%S 1,1,2,1,4,8,10,8,4,32,64,116,136,116,64,32,384,768,1800,2448,2844,

%T 2448,1800,768,384,6144,12288,35200,51968,75232,79424,75232,51968,

%U 35200,12288,6144,122880,245760,830720,1292800,2226640,2652320,3037720

%N A triangle sequence from polynomial coefficients: p(x, n) = (4*(n - 1) + (4(n - 1))*x^2)*p(x, n - 1) + 2*(n - 1)x^2*p(x, n - 2).

%C Row sums are:

%C {1, 4, 34, 560, 13644, 441088, 17779960, 858731136, 48337863056,

%C 3107362933760, 224600212765728,...}.

%F p(x, n) = (4*(n - 1) + (4(n - 1))*x^2)*p(x, n - 1) + 2*(n - 1)x^2*p(x, n - 2);

%F t(n,m)=coefficients(p(x,n)).

%e {1},

%e {1, 2, 1},

%e {4, 8, 10, 8, 4},

%e {32, 64, 116, 136, 116, 64, 32},

%e {384, 768, 1800, 2448, 2844, 2448, 1800, 768, 384},

%e {6144, 12288, 35200, 51968, 75232, 79424, 75232, 51968, 35200, 12288, 6144},

%e {122880, 245760, 830720, 1292800, 2226640, 2652320, 3037720, 2652320, 2226640, 1292800, 830720, 245760, 122880},

%e {2949120, 5898240, 22960128, 37072896, 73799040, 95306496, 127247424, 128264448, 127247424, 95306496, 73799040, 37072896, 22960128, 5898240, 2949120},

%e {82575360, 165150720, 727179264, 1206632448, 2720886784, 3724722176, 5660473952, 6297118912, 7168383824, 6297118912, 5660473952, 3724722176, 2720886784, 1206632448, 727179264, 165150720, 82575360}, {2642411520, 5284823040, 25959333888, 43991433216, 110705475584, 158396514304, 269384328192, 322223818752, 412559407616, 405067841536, 412559407616, 322223818752, 269384328192, 158396514304, 110705475584, 43991433216, 25959333888, 5284823040, 2642411520}, {95126814720, 190253629440, 1031149191168, 1776917938176, 4933022367744, 7307685494784, 13732208898048, 17369376989184, 24651863020224, 26295847910784, 29833308257184, 26295847910784, 24651863020224, 17369376989184, 13732208898048, 7307685494784, 4933022367744, 1776917938176, 1031149191168, 190253629440, 95126814720}

%t Clear[p, x];

%t p[x, 0] = 1; p[x, 1] = x^2 + 2*x + 1;

%t p[x_, n_] := p[x, n] = (4*(n - 1) + (4(n - 1))*x^2)*p[x, n - 1] + 2*(n - 1)x^2*p[x, n - 2];

%t Flatten[Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]]

%K nonn,tabl

%O 0,3

%A _Roger L. Bagula_ and _Gary W. Adamson_, Feb 16 2009

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