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%I #13 May 14 2016 12:23:59
%S 1,1,7,1,1,45,96,20,1,168,1316,1730,420,10,1,481,9486,35959,35959,
%T 9486,481,1,1,1165,48103,395692,974863,816025,226765,17248,196,1,2507,
%U 193830,2977338,14467635,26621385,19654740,5661084,558168,13496,28,1,4935
%N Triangle T(n,k) read by rows: the coefficient [x^k] of the series (1-x)^(2n-1)*Sum_{l>=0} A001263(n+3*l,3*l+1)*x^l, in row n>=1 with exponents k>=0.
%C Row sums are 1, 9, 162, 3645, 91854, 2480058, 70150212, 2051893701, 61556811030,
%C 1883638417518,....
%F Let A(n,m) = A001263(n+m,m+1), then T(n,k) = [x^k] (1-x)^(2n-1) * Sum_{l>=0} A(n,3l) *x^l.
%F If n == 2 (mod 3), then T(n,k) = T(n,5(n-2)/3+2-k).
%e 1;
%e 1, 7, 1;
%e 1, 45, 96, 20;
%e 1, 168, 1316, 1730, 420, 10;
%e 1, 481, 9486, 35959, 35959, 9486, 481, 1;
%e 1, 1165, 48103, 395692, 974863, 816025, 226765, 17248, 196;
%e 1, 2507, 193830, 2977338, 14467635, 26621385, 19654740, 5661084, 558168, 13496, 28;
%e 1, 4935, 662019, 17287475, 146146455, 494277453, 735137025, 494277453, 146146455, 17287475, 662019, 4935, 1;
%e 1, 9058, 1993006, 82927275, 1124706060, 6284964498, 16252722579, 20538756768, 12843595065, 3874383980, 524761304, 27575207, 415404, 825;
%t p[x_, n_] = (1 - x)^(2*n - 1)*Sum[(Binomial[3*k + n, 3* k] Binomial[3*k + n, 1 + 3*k]/(3*k + n))*x^k, {k, 0, Infinity}];
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
%t Flatten[%]
%Y Cf. A001263, A034839
%K nonn,tabf
%O 1,3
%A _Roger L. Bagula_, Jun 01 2010
%E Edited by _R. J. Mathar_, May 13 2016
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