A203709 -id:A203709 - cp's OEIS frontend

A203716 E.g.f.: Product_{n>=1} (exp(2*x^n) + 1)/2.

1, 1, 4, 16, 104, 696, 6272, 57856, 652416, 7657600, 104244992, 1475430144, 23426373632, 387521615872, 7034561925120, 132850810138624, 2709375373672448, 57456525327335424, 1301169515685085184, 30573796812553584640, 760486440376336908288, 19600568102376899608576

Offset: 0

Paul D. Hanna, Jan 04 2012

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 104*x^4/4! + 696*x^5/5! +...
where the e.g.f. equals the product:
A(x) = (exp(2*x)+1)/2 * (exp(2*x^2)+1)/2 * (exp(2*x^3)+1)/2 * (exp(2*x^4)+1)/2 *...
The log of the e.g.f. begins:
log(A(x)) = x + 3*x^2/2! + x^3 + 34*x^4/4! + x^5 + 1096*x^6/6! + x^7 + 56848*x^8/8! + x^9 +...+ A203715(n)*x^n/n! +...
Note that the coefficients of the odd powers of x in log(A(x)) equals 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A203715 (log), A203709, A270664, A270665, A270666.

nmax = 25; Range[0, nmax]! * CoefficientList[Series[Product[1/(1 - Tanh[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 21 2016 *)

{a(n)=n!*polcoeff(prod(k=1, n, (exp(2*x^k+x*O(x^n))+1)/2), n)}

A203715 E.g.f.: Sum_{n>=1} log((1 + exp(2*x^n))/2).

Original entry on oeis.org

1, 3, 6, 34, 120, 1096, 5040, 56848, 362880, 5451136, 39916800, 688876288, 6227020800, 130789805056, 1307674368000, 29497569445888, 355687428096000, 9746045395173376, 121645100408832000, 3451902721622867968, 51090942171709440000, 1686006043164464644096

Offset: 1

Paul D. Hanna, Jan 04 2012

nonn

			E.g.f.: A(x) = x + 3*x^2/2! + x^3 + 34*x^4/4! + x^5 + 1096*x^6/6! + x^7 + 56848*x^8/8! + x^9 + 5451136*x^10/10! + x^11 +...
where A(x) = log((1+exp(2*x))/2) + log((1+exp(2*x^2))/2) + log((1+exp(2*x^3))/2) + log((1+exp(2*x^4))/2) +...
The exponentiation of the e.g.f. begins:
exp(A(x)) = 1 + x + 4*x^2/2! + 16*x^3/3! + 104*x^4/4! + 696*x^5/5! + 6272*x^6/6! + 57856*x^7/7! + 652416*x^8/8! +...+ A203716(n)*x^n/n! +...

Cf. A203709, A203716 (exp).

nmax = 25; Rest[Range[0, nmax]! * CoefficientList[Series[Sum[Log[1/(1 - Tanh[x^k])], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Mar 21 2016 *)

{a(n)=n!*polcoeff(sum(m=1,n,log((1+exp(2*x^m+x*O(x^n)))/2)),n)}

a(2*n-1) = (2*n-1)!.

cp's OEIS Frontend

A203716 E.g.f.: Product_{n>=1} (exp(2*x^n) + 1)/2.

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