A193281 E.g.f.: A(x) = 1/[(1 - x^2)^(1 + 1/x)].

1, 1, 3, 10, 49, 271, 1861, 14218, 125945, 1222525, 13274551, 155993586, 2006410033, 27654327715, 411146102641, 6498852124066, 109654601938801, 1954603251773273, 36899871955766155, 732088187170903162, 15288727416236948241, 334089096072180624631

Offset

0,3

Comments

More generally, we have the identity:

Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=x.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

E.g.f.: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + k*x).

E.g.f.: A(x)=1/[ (1 - x^2)*(1 - x^2)^(1/x)] = 1 + x*(1+x)/(G(0) - x*(1+x)) ; G(k) = 1 + k + x*(k*x+x+1) - x*(k+1)*(k*x+2*x+1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 02 2011

a(n) ~ n! * (n + log(n) + 2 + gamma - log(2))/4, where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 08 2013

From Peter Bala, Aug 31 2014: (Start)

E.g.f.: exp( sum{n = 1..inf} x^n/A110654(n) ).

It appears that a(n) = 1 ( mod n*(n - 1) ) for n even, while a(n) = 1 ( mod n*(n - 1)/2 ) for n odd [checked up to a(100)]. (End)

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k/A110654(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 30 2022

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 49*x^4/4! + 271*x^5/5! +...
where:
A(x)^(x/(1+x)) = 1 + x^2 + x^4 + x^6 + x^8 + x^10 + x^12 +...
Also,
A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+2*x)/2! + x^3*(1+x)*(1+2*x)*(1+3*x)/3! + x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/4! +...
The logarithm begins:
log(A(x)) = x + x^2 + x^3/2 + x^4/2 + x^5/3 + x^6/3 + x^7/4 + x^8/4 +...

Mathematica

CoefficientList[Series[1/((1-x^2)^(1+1/x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(1/(1 - x^2 +x^2*O(x^n))^((1+x)/x), n)}
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, x^m/m!*prod(k=1, m, 1+k*x+x*O(x^n))), n)}
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j/((j+1)\2)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Apr 30 2022

Crossrefs

Cf. A110654, A193287, A193288, A193289, A193290, A246689.

Sequence in context: A220362 A121950 A005921 * A367754 A143921 A082426

Adjacent sequences: A193278 A193279 A193280 * A193282 A193283 A193284

Keyword

nonn

Author

Paul D. Hanna, Jul 21 2011

Status

approved