A381171
Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + x*cosh(x)) ).
Original entry on oeis.org
1, 1, 2, 9, 72, 725, 8640, 124117, 2117248, 41477193, 913305600, 22371549761, 604476094464, 17858943664861, 572524035586048, 19793963392789965, 734249332747960320, 29090332675789113617, 1225991945551031304192, 54765451909152748484857, 2584803582762012599910400
Offset: 0
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a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+1, k)*a185951(n, k))/(n+1);
A215364
E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*cosh(x*A(x)).
Original entry on oeis.org
1, 1, 4, 33, 408, 6725, 139200, 3475717, 101722880, 3416079753, 129507425280, 5471712276041, 254965505507328, 12990483544072333, 718474796305989632, 42871067358096134445, 2745230569464318197760, 187780115708775158008337, 13665196427126843296972800
Offset: 0
E.g.f: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! + 6725*x^5/5! +...
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CoefficientList[1/x*InverseSeries[Series[x-x^2*Cosh[x], {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 13 2014 *)
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{a(n)=n!*polcoeff(1/(1 - x*cosh(x+x*O(x^n)))^(n+1)/(n+1),n)}
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{a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*cosh(x+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
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a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+k+1,k)/(n+k+1)*a185951(n, k)); \\ Seiichi Manyama, Feb 16 2025
A381172
E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cosh(x * A(x)) ).
Original entry on oeis.org
1, 1, 6, 75, 1416, 36065, 1160400, 45182347, 2066343552, 108594342369, 6449557524480, 427226389872491, 31230489190382592, 2497416890105693569, 216875134620623990784, 20324880119519860657515, 2044641793664946681446400, 219762483007148574205773377, 25134006030221243013604835328
Offset: 0
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a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+2*k+1, k)/(n+2*k+1)*a185951(n, k));
Showing 1-3 of 3 results.
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