A371044
E.g.f. satisfies A(x) = 1 + x^3*exp(x*A(x)).
Original entry on oeis.org
1, 0, 0, 6, 24, 60, 120, 5250, 80976, 726264, 4839120, 86487390, 2283242280, 42585905076, 590667519624, 10115535833130, 286758920451360, 8128299117822960, 186279550983756576, 4123388294626654134, 118916807955913504440, 4102548791571529697580
Offset: 0
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nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^4]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 10 2024 *)
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a(n) = n!*sum(k=0, n\3, k^(n-3*k)*binomial(n-3*k+1, k)/((n-3*k+1)*(n-3*k)!));
A371018
Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^2*exp(x)) ).
Original entry on oeis.org
1, 0, 2, 6, 60, 620, 7950, 129402, 2365496, 50512968, 1208642490, 32223422990, 947694971652, 30435132773916, 1061061668979494, 39889366397571810, 1608910488000292080, 69305890226183224592, 3175519952912430375666, 154216789672147809137046
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x^2*exp(x)))/x))
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a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(n+1, k)/(n-2*k)!)/(n+1);
A372019
G.f. A(x) satisfies A(x) = ( 1 + 9*x*A(x)/(1 - x*A(x)) )^(1/3).
Original entry on oeis.org
1, 3, 3, 3, 30, 57, 84, 867, 1893, 3162, 33132, 76953, 136812, 1446204, 3478764, 6420387, 68260134, 167946159, 317782524, 3392340186, 8479140510, 16332164868, 174873206424, 442212416121, 863222622780, 9264327739716, 23637757714788, 46624054987452
Offset: 0
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A371019 := proc(n)
add(9^k*binomial((n+1)/3,k)*binomial(n-1,k-1),k=0..n) ;
%/(n+1) ;
end proc:
seq(A371019(n),n=0..60) ; # R. J. Mathar, Apr 22 2024
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a(n) = sum(k=0, n, 9^k*binomial(n/3+1/3, k)*binomial(n-1, n-k))/(n+1);
A371045
E.g.f. satisfies A(x) = 1 + x^3*A(x)*exp(x*A(x)).
Original entry on oeis.org
1, 0, 0, 6, 24, 60, 840, 15330, 161616, 1572984, 29031120, 636008670, 11426850600, 210095235636, 5137568918664, 139255673359530, 3574532174656800, 95923063388359920, 2974073508961556256, 98747639807081454774, 3287535337205171488440
Offset: 0
A371046
E.g.f. satisfies A(x) = 1 + x^3*A(x)^2*exp(x*A(x)).
Original entry on oeis.org
1, 0, 0, 6, 24, 60, 1560, 25410, 242256, 3508344, 85882320, 1724406750, 32784999720, 839182482996, 24162605028744, 659439484706730, 19415319297457440, 658935736181053680, 23245444335085544736, 835819877947421773494, 32462532011236141677240
Offset: 0
A371021
Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^3/6*exp(x)) ).
Original entry on oeis.org
1, 0, 0, 1, 4, 10, 80, 1015, 9016, 80724, 1092120, 16872405, 246966940, 3932454526, 73869476044, 1485097614455, 30688224287280, 682450482838440, 16508839426673136, 420562937260614249, 11193327347979937140, 315276822746559147250, 9383980947735649740100
Offset: 0
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Join[{1}, Table[(n!/(n + 1))*Sum[k^(n - 3*k)*Binomial[n + 1, k]/(6^k*(n - 3*k)!), {k, 0, Floor[n/3]}], {n, 30}]] (* Wesley Ivan Hurt, Aug 05 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x^3/6*exp(x)))/x))
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a(n) = n!*sum(k=0, n\3, k^(n-3*k)*binomial(n+1, k)/(6^k*(n-3*k)!))/(n+1);
Showing 1-6 of 6 results.