A221370 O.g.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (k + x) / (1 + k^2*x + k*x^2).

1, 1, 4, 21, 183, 2362, 42449, 1012897, 30961412, 1179154241, 54727128731, 3040047461530, 199109235070645, 15182265283487213, 1333242114217704924, 133577535961042535669, 15144191953510005439455, 1928873660857769308675146, 274228718414760130917382185

Offset

0,3

Comments

Compare to the identity:

Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + x) / (1 + k*x + k*x^2) = 1/(1-x-x^2).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..208

Formula

O.g.f.: 1/(1-x*(1+x)/(1-1*2*x/(1-2*x*(2+x)/(1-2*3*x/(1-3*x*(3+x)/(1-3*4*x/(1-4*x*(4+x)/(1-4*5*x/(1-5*x*(5+x)/(1-5*6*x/(1+...))))))))))) (continued fraction).

a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Nov 02 2014

Example

O.g.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 183*x^4 + 2362*x^5 + 42449*x^6 + ...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!*x^2*(1+x)*(2+x)/((1+x+x^2)*(1+4*x+2*x^2)) + 3!*x^3*(1+x)*(2+x)*(3+x)/((1+x+x^2)*(1+4*x+2*x^2)*(1+9*x+3*x^2)) + 4!*x^4*(1+x)*(2+x)*(3+x)*(4+x)/((1+x+x^2)*(1+4*x+2*x^2)*(1+9*x+3*x^2)*(1+16*x+4*x^2)) + ...

Mathematica

a[n_] := Coefficient[Sum[m!*x^m*Product[(k + x)/(1 + k^2*x + k*x^2), {k, 1, m}], {m, 0, n}] + O[x]^(n + 1), x, n]; Table[a[n], {n, 0, 18}] (* Robert P. P. McKone, Sep 15 2023 *)

Prog

(PARI) {a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (k+x)/(1+k^2*x+k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A221371, A208237, A210438.

Sequence in context: A230682 A231220 A231434 * A224500 A158108 A158258

Adjacent sequences: A221367 A221368 A221369 * A221371 A221372 A221373

Keyword

nonn

Author

Paul D. Hanna, Jan 13 2013

Status

approved