A380515
Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217
Offset: 0
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a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));
A380516
Expansion of e.g.f. exp(x*G(x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 9, 157, 4129, 146001, 6502681, 349790029, 22069858497, 1598577634369, 130757736096361, 11922399644742621, 1199121973234651489, 131887738425602277457, 15748194681225620534649, 2028885239529647188594381, 280525944581514367875035521, 41434950383158772951280658689
Offset: 0
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Join[{1}, Table[(n-1)! * LaguerreL[n-1, 3*n+1, -1], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 26 2025 *)
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a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 3*n+1, -1));
A380514
Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 5, 67, 1537, 50021, 2107021, 108885295, 6665443457, 471522589417, 37843890892021, 3397250515809371, 337267132243022785, 36687625652474612557, 4339368321317331858557, 554467482301151809302151, 76112537023512618262963201, 11170667360636927554290623825, 1745500813880455301486766050917
Offset: 0
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a(n) = if(n==0, 1, 2*n!*sum(k=0, n-1, binomial(2*n+2*k, k)/((2*n+2*k)*(n-k-1)!)));
A382031
E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x)^2)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 3, 43, 1177, 46681, 2419291, 154587427, 11735209585, 1031418915121, 102979800567091, 11510663862332251, 1423811747933017609, 193073662118499898633, 28479005472094048953355, 4539456019668776334683731, 777538096585429376795405281, 142419954152382631361835929185
Offset: 0
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a(n) = if(n==0, 1, n!*sum(k=0, n-1, (2*k+1)^(n-k-1)*binomial(n+3*k, k)/((n+3*k)*(n-k-1)!)));
A382016
E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 3, 37, 901, 32141, 1502701, 86737645, 5952271977, 473117681881, 42731313784921, 4321503662185601, 483709266378568429, 59360036142346311685, 7924411424305558028757, 1143251381667547987358581, 177245340974472998607370321, 29386977237154379581209716657
Offset: 0
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a(n) = if(n==0, 1, n!*sum(k=0, n-1, (k+1)^(n-k-1)*binomial(n+3*k, k)/((n+3*k)*(n-k-1)!)));
Showing 1-5 of 5 results.