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A171502 Coefficients of polynomials related to SU(n) as polynomial product with powers n^2-1: p(x,n)=p(x,n-1)*sum[x^i,{i,0,2*n-1}] 0

%I

%S 1,1,1,1,1,1,2,3,4,4,4,3,2,1,1,3,6,10,14,18,21,23,23,21,18,14,10,6,3,

%T 1,1,4,10,20,34,52,73,96,119,140,157,168,172,168,157,140,119,96,73,52,

%U 34,20,10,4,1,1,5,15,35,69,121,194,290,409,549,706,874,1045,1209,1356,1476

%N Coefficients of polynomials related to SU(n) as polynomial product with powers n^2-1: p(x,n)=p(x,n-1)*sum[x^i,{i,0,2*n-1}]

%C These polynomials are similar to the A142724 Poincaré polynomials that inspired them.

%C Row sums are;

%C {1, 4, 24, 192, 1920, 23040, 322560, 5160960, 92897280, 1857945600,...}.

%C Coefficient lengths as powers are A005563:

%C Table[Length[CoefficientList[p[x, n], x]] - 1, {n, 1, 10}]

%C {0, 3, 8, 15, 24, 35, 48, 63, 80, 99,...};

%C a(n)=a(n-1)+2*n-1

%D Samuel I. Goldberg, Curvature and Homology, Dover, New York, 1998, page 144.

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000059">The inversion number of a standard Young tableau as defined by Haglund and Stevens.</a>

%F p(x,n)=p(x,n-1)*sum[x^i,{i,0,2*n-1}]

%e {1},

%e {1, 1, 1, 1},

%e {1, 2, 3, 4, 4, 4, 3, 2, 1},

%e {1, 3, 6, 10, 14, 18, 21, 23, 23, 21, 18, 14, 10, 6, 3, 1},

%e {1, 4, 10, 20, 34, 52, 73, 96, 119, 140, 157, 168, 172, 168, 157, 140, 119, 96, 73, 52, 34, 20, 10, 4, 1},

%e {1, 5, 15, 35, 69, 121, 194, 290, 409, 549, 706, 874, 1045, 1209, 1356, 1476, 1561, 1605, 1605, 1561, 1476, 1356, 1209, 1045, 874, 706, 549, 409, 290, 194, 121, 69, 35, 15, 5, 1}, {1, 6, 21, 56, 125, 246, 440, 730, 1139, 1688, 2394, 3268, 4313, 5522, 6877, 8348, 9894, 11464, 13000, 14440, 15722, 16788, 17588, 18084, 18252, 18084, 17588, 16788, 15722, 14440, 13000, 11464, 9894, 8348, 6877, 5522, 4313, 3268, 2394, 1688, 1139, 730, 440, 246, 125, 56, 21, 6, 1},

%e {1, 7, 28, 84, 209, 455, 895, 1625, 2764, 4452, 6846, 10114, 14427, 19949, 26826, 35174, 45067, 56525, 69504, 83888, 99485, 116027, 133175, 150529, 167642, 184038, 199232, 212752, 224161, 233079, 239202, 242318, 242318, 239202, 233079, 224161, 212752, 199232, 184038, 167642, 150529, 133175, 116027, 99485, 83888, 69504, 56525, 45067, 35174, 26826, 19949, 14427, 10114, 6846, 4452, 2764, 1625, 895, 455, 209, 84, 28, 7, 1}

%t Clear[p, x, n, a]

%t p[x, 1] = 1; p[x, 2] = x^3 + x^2 + x + 1;

%t p[x_, n_] := p[x, n] = p[x, n - 1]*Sum[x^i, {i, 0, 2*n - 1}]

%t a = Table[CoefficientList[p[x, n], x], {n, 1, 10}]

%t Flatten[a]

%Y Cf. A142724.

%K nonn,tabf,uned

%O 1,7

%A _Roger L. Bagula_, Dec 10 2009

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Last modified October 14 00:30 EDT 2019. Contains 327991 sequences. (Running on oeis4.)