A229286 G.f.: Sum_{n>=0} x^n / Product_{k=1..2*n} (1 - k*x).

1, 1, 4, 18, 102, 684, 5216, 44388, 415672, 4234904, 46525992, 547327904, 6854491840, 90940138256, 1272862982272, 18728235407712, 288765445378272, 4653013453323968, 78164063007644288, 1365903793778043712, 24781386644286473856, 465969812835308934272, 9066115469486822859392

Offset

0,3

Comments

Compare to o.g.f. of Bell numbers: Sum_{n>=0} x^n / Product_{k=0..n} (1 - k*x).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..160

Ira M. Gessel, General case of the some R-recursions, answer to question on MathOverflow (2024).

Formula

a(n) = R(n-1, 0) for n > 0 with a(0) = 1 where R(n, q) = (4*q+3)*R(n-1, q) - 2*(q+1)*(2*q+1)*R(n-2, q) + R(n-1, q+1) for n > 0, q >= 0 with R(n, q) = [n = 0] for n < 1, q >= 0. - Mikhail Kurkov, Oct 02 2024

Example

G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 102*x^4 + 684*x^5 + 5216*x^6 +...
where
A(x) = 1 + x/((1-x)*(1-2*x)) + x^2/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)) +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=0, n, x^m/prod(k=1, 2*m, 1-k*x+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) upto(n) = my(v1, v2, v3, v4); v1 = vector(n, i, 1); v2 = vector(n, i, 0); v3 = vector(n+1, i, 0); for(i=1, min(n+1, 2), v3[i] = 1); for(i=1, n-1, v4 = v1; for(j=1, n-i, v1[j] = (4*j-1)*v1[j] - 2*j*(2*j-1)*v2[j] + v1[j+1]); v3[i+2] = v1[1]; v2 = v4); v3 \\ Mikhail Kurkov, Oct 02 2024

Crossrefs

Cf. A229285.

Sequence in context: A327833 A350267 A064852 * A191365 A335459 A159666

Adjacent sequences: A229283 A229284 A229285 * A229287 A229288 A229289

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2013

Status

approved