A381267 a(n) = numerator( [(x*y*z*u)^n] 1/sqrt(1 - (x + y + z + u*(y + z))) ).
1, 15, 31185, 6381375, 409933148625, 115551955934415, 561860686475913825, 179982394552964750175, 245527483089290688069980625, 84259935283701238220954169375, 473788223464393905637179153328785, 169752647693877043154936308907932575, 15821279983229628402902553309640505635425
Offset: 0
Links
- S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 24.
Programs
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Mathematica
a[n_]:=Numerator[SeriesCoefficient[1/Sqrt[1-(x+y+z+u(y+z))],{x,0,n},{y,0,n},{z,0,n},{u,0,n}]]; Array[a,13,0]
Formula
a(n) = numerator( [x^n] hypergeom( [1/2, 1/6, 1/2, 5/6], [1, 1, 1], 108*x) ).
a(n) = numerator( 2^(2*n-1) * 27^n * Gamma(n+1/6) * Gamma(n+1/2)^2 * Gamma(n+5/6)/(Pi^2 * (n!)^4) ).