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

1, 1, 2, 2, 6, 12, 6, 1, 26, 73, 72, 24, 12, 156, 516, 732, 480, 120, 2, 126, 1206, 4322, 7680, 7320, 3600, 720, 52, 1408, 11352, 42448, 87652, 106800, 76800, 30240, 5040, 11, 992, 17406, 125444, 480731, 1103460, 1601148, 1486800, 859320, 282240, 40320

Offset: 0

Author

N. J. A. Sloane, Jan 20 2017

Keywords

• nonn
• tabf

Examples

			Triangle begins:
1,
1,
2,2,
6,12,6,
1,26,73,72,24,
12,156,516,732,480,120,
2,126,1206,4322,7680,7320,3600,720,
...
		

Links

• Mark Dukes, Chris D White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016. See Fig. 6 p. 14.
• Mark Dukes, Chris D White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45 See Fig. 6 p. 17.

Crossrefs

Cf. A269722.

Programs

Mathematica

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]&];
Table[row[n], {n, 0, 8}] // Flatten (* from PARI *)

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))))))));
tabf(nn) = for (n=0, nn, rown = vL(n); for (k=1, #rown, print1(rown[k], ", ")); print()); \\ Michel Marcus, Jan 21 2017

Formula

See Theorem 23 in the Dukes link.

Extensions

More terms from Michel Marcus, Jan 21 2017