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A281351 Irregular triangle read by rows: coefficients of polynomials arising in calculation of squares of certain web-coloring matrices.

%I A281351 #16 Sep 25 2018 04:06:13
%S A281351 1,1,2,2,6,12,6,1,26,73,72,24,12,156,516,732,480,120,2,126,1206,4322,
%T A281351 7680,7320,3600,720,52,1408,11352,42448,87652,106800,76800,30240,5040,
%U A281351 11,992,17406,125444,480731,1103460,1601148,1486800,859320,282240,40320
%N A281351 Irregular triangle read by rows: coefficients of polynomials arising in calculation of squares of certain web-coloring matrices.
%H A281351 Mark Dukes, Chris D White, <a href="http://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016. See Fig. 6 p. 14.
%H A281351 Mark Dukes, Chris D White, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p45">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45 See Fig. 6 p. 17.
%F A281351 See Theorem 23 in the Dukes link.
%e A281351 Triangle begins:
%e A281351 1,
%e A281351 1,
%e A281351 2,2,
%e A281351 6,12,6,
%e A281351 1,26,73,72,24,
%e A281351 12,156,516,732,480,120,
%e A281351 2,126,1206,4322,7680,7320,3600,720,
%e A281351 ...
%t A281351 row[n_] := If[n<2, {1}, Sum[x^m*Sum[(-1)^(m-b-c) Binomial[j, b] Binomial[m-j, c] Binomial[b c, n], {c, 0, m-j}], {m, 2, 2n}, {j, 1, m-1}, {b, 0, j}] // DeleteCases[CoefficientList[#, x], 0]&];
%t A281351 Table[row[n], {n, 0, 8}] // Flatten (* from PARI *)
%o A281351 (PARI) vL(n) = if (n==0, [1], select(x->x, Vecrev(sum(m=2, 2*n, x^m*sum(j=1, m-1, sum(b=0, j, sum(c=0, m-j, (-1)^(m-b-c)*binomial(j,b)*binomial(m-j,c)*binomial(b*c, n))))))));
%o A281351 tabf(nn) = for (n=0, nn, rown = vL(n); for (k=1, #rown, print1(rown[k], ", ")); print()); \\ _Michel Marcus_, Jan 21 2017
%Y A281351 Cf. A269722.
%K A281351 nonn,tabf
%O A281351 0,3
%A A281351 _N. J. A. Sloane_, Jan 20 2017
%E A281351 More terms from _Michel Marcus_, Jan 21 2017