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);
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));
A215363
E.g.f. satisfies: A(x) = x + A(x)^2*cosh(A(x)).
Original entry on oeis.org
1, 2, 12, 132, 2040, 40350, 974400, 27805736, 915505920, 34160797530, 1424581678080, 65660547312492, 3314551571595264, 181866769617012662, 10777121944589844480, 685937077729538151120, 46668919680893409361920, 3380042082757952844150066, 259638732115410022642483200
Offset: 1
E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
Series expressions:
A(x) = x + cosh(x)*x^2 + d/dx cosh(x)^2*x^4/2! + d^2/dx^2 cosh(x)^3*x^6/3! + d^3/dx^3 cosh(x)^4*x^8/4! +...
log(A(x)/x) = cosh(x)*x + d/dx cosh(x)^2*x^3/2! + d^2/dx^2 cosh(x)^3*x^5/3! + d^3/dx^3 cosh(x)^4*x^7/4! +...
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Rest[CoefficientList[InverseSeries[Series[x - x^2*Cosh[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)
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{a(n)=n!*polcoeff(serreverse(x-x^2*cosh(x+x*O(x^n))), n)}
for(n=1, 25, print1(a(n), ", "))
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{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^m*x^(2*m)/m!)); n!*polcoeff(A, n)}
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{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^m*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
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{a(n)=n!*polcoeff(x/(1 - x*cosh(x+x*O(x^n)))^n/n, n)}
Showing 1-3 of 3 results.
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