A364981
E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^3).
Original entry on oeis.org
1, 1, 4, 39, 580, 11685, 298566, 9248701, 336886936, 14112113049, 668422303210, 35325208755441, 2060811941835780, 131547166492534117, 9120279070776381886, 682489450793082237285, 54828316394224735284016, 4706545644403274325580593
Offset: 0
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Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[3*n-2*k+1,k] / ((3*n-2*k+1)*(n-k)!), {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)
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a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*n-2*k+1, k)/((3*n-2*k+1)*(n-k)!));
A365030
E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^3).
Original entry on oeis.org
1, 1, 7, 55, 709, 11761, 243181, 6054763, 175803097, 5847578785, 219175994521, 9144024668131, 420340277237365, 21111584238219697, 1150333949592549541, 67589878866533749531, 4260172601206280708401, 286737199114729515029569
Offset: 0
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Array[#!*Sum[ (# - k + 1)^(k - 1)*Binomial[3 k, # - k]/k!, {k, 0, #}] &, 18, 0] (* Michael De Vlieger, Aug 18 2023 *)
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a(n) = n!*sum(k=0, n, (n-k+1)^(k-1)*binomial(3*k, n-k)/k!);
A372201
E.g.f. A(x) satisfies A(x) = exp( 3 * x / (1 - x * A(x)^(1/3))^3 ).
Original entry on oeis.org
1, 3, 27, 351, 6309, 145143, 4083669, 136159299, 5256248265, 230783968395, 11364265672929, 620524946670687, 37222254648712989, 2433741005377774719, 172301622840992025117, 13133140607475128862747, 1072406955985984437773841, 93406430850089038192704915
Offset: 0
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a(n, r=3, s=3, t=0, u=1) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(n+(s-1)*k-1, n-k)/k!);
A382059
E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 7, 127, 3733, 152161, 7939261, 505087843, 37920697753, 3281899787137, 321700411900441, 35227497466867531, 4262151791317099285, 564639582580738851265, 81290104199287214904037, 12637400195063381931755731, 2109868901338065949399370161, 376504852688521502050554789889
Offset: 0
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a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, (k+1)^(n-k-1)*binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));
Showing 1-4 of 4 results.