A358014
Expansion of e.g.f. 1/(1 - x^3 * (exp(x) - 1)).
Original entry on oeis.org
1, 0, 0, 0, 24, 60, 120, 210, 40656, 363384, 2117520, 9980190, 520250280, 9496208436, 109522054824, 982593614730, 28426015541280, 762523155318000, 14192088961120416, 204618562767970614, 4906638448867994040, 154037798077765359660, 4000484484370905087480
Offset: 0
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With[{nn=30},CoefficientList[Series[1/(1-x^3 (Exp[x]-1)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 26 2024 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*(exp(x)-1))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=4, i, 1/(j-3)!*v[i-j+1]/(i-j)!)); v;
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a(n) = n!*sum(k=0, n\4, k!*stirling(n-3*k, k, 2)/(n-3*k)!);
A370989
Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2*(exp(x) - 1)) ).
Original entry on oeis.org
1, 0, 0, 6, 12, 20, 2910, 22722, 117656, 8482392, 143398170, 1519998590, 79655138772, 2206506673956, 39101112995126, 1798446230741370, 68667380639283120, 1795441154500375472, 81344029377887798706, 3830461514154681289974, 135388937631209203030700
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2*(exp(x)-1)))/x))
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a(n) = sum(k=0, n\3, (n+k)!*stirling(n-2*k, k, 2)/(n-2*k)!)/(n+1);
A375662
Expansion of e.g.f. 1 / (1 - x^2 * (exp(x) - 1))^2.
Original entry on oeis.org
1, 0, 0, 12, 24, 40, 2220, 15204, 70672, 1723824, 22710420, 202577980, 3841065624, 71221859592, 994632663388, 19005155049300, 421055077585440, 8033764197776224, 172109549363348772, 4285658639255113836, 101794836650015825320, 2516190299149752959160
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*(exp(x)-1))^2))
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a(n) = n!*sum(k=0, n\3, (k+1)!*stirling(n-2*k, k, 2)/(n-2*k)!);
A375663
Expansion of e.g.f. 1 / (1 - x^2 * (exp(x) - 1))^3.
Original entry on oeis.org
1, 0, 0, 18, 36, 60, 4410, 30366, 141288, 4173336, 56307150, 504947850, 10795641516, 209176625268, 2958760573314, 60807476490390, 1419440085948240, 27655117897680816, 621153635750802198, 16250306025184563330, 396542042830732066260, 10152940698142734694860
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*(exp(x)-1))^3))
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a(n) = n!*sum(k=0, n\3, (k+2)!*stirling(n-2*k, k, 2)/(n-2*k)!)/2;
A371304
E.g.f. satisfies A(x) = 1/(1 - x^2*(exp(x*A(x)) - 1)).
Original entry on oeis.org
1, 0, 0, 6, 12, 20, 1470, 12642, 70616, 2131992, 39352410, 470186750, 11032124532, 295053244356, 5896487364950, 146264289411450, 4625791393554480, 130492119237611312, 3837833086814864946, 135471306780659593206, 4800394977109827314060
Offset: 0
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a(n) = n!*sum(k=0, n\3, (n-k)!*stirling(n-2*k, k, 2)/((n-2*k)!*(n-2*k+1)!));
A375715
Expansion of e.g.f. 1 / sqrt(1 - x^2 * (exp(x) - 1)).
Original entry on oeis.org
1, 0, 0, 3, 6, 10, 285, 1911, 8848, 147456, 1818225, 15966775, 244374636, 4105980528, 55574016589, 938220142965, 18765940185840, 342231152117536, 6765035069902833, 154060159512672315, 3469311695227952260, 80672955862303202160, 2068943441492081794101
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-x^2*(exp(x)-1))))
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a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = n!*sum(k=0, n\3, a001147(k)*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));
Showing 1-6 of 6 results.