A360239 G.f. A(x) = exp( Sum_{k>=1} A360238(k) * x^k/k ), where A360238(k) = [y^k*x^k/k] log( Sum_{m>=0} (m + y)^(2*m) * x^m ) for k >= 1.

1, 2, 23, 502, 16414, 716936, 39167817, 2567058766, 196159319943, 17118727499178, 1679643875717867, 183020512751712554, 21928106267349661127, 2865208654370111795940, 405479888251812823615679, 61785441098476295018209264, 10085622916281496742096639996

Offset

0,2

Comments

Related series: C(x)^2 = exp( Sum_{k>=1} A000984(k) * x^k/k ), where C(x) = 1 + x*C(x)^2 is the Catalan function and A000984(k) = [y^k*x^k/k] log( Sum_{m>=0} (1 + y)^(2*m) * x^m ) for k >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

a(n) ~ (1 - exp(-1)/4) * 2^(2*n) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 12 2023

Example

G.f.: A(x) = 1 + 2*x + 23*x^2 + 502*x^3 + 16414*x^4 + 716936*x^5 + 39167817*x^6 + 2567058766*x^7 + 196159319943*x^8 + 17118727499178*x^9 + ...
such that
log(A(x)) = 2*x + 42*x^2/2 + 1376*x^3/3 + 60934*x^4/4 + 3377252*x^5/5 + 224036904*x^6/6 + 17282039280*x^7/7 + ... + A360238(n)*x^n/n + ...
where A360238(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (m + y)^(2*m) * x^m ) = (y^2 + 2*y + 1)*x + (y^4 + 12*y^3 + 42*y^2 + 60*y + 31)*x^2/2 + (y^6 + 30*y^5 + 297*y^4 + 1376*y^3 + 3348*y^2 + 4188*y + 2140)*x^3/3 + (y^8 + 56*y^7 + 1100*y^6 + 10792*y^5 + 60934*y^4 + 209464*y^3 + 436692*y^2 + 510952*y + 258779)*x^4/4 + (y^10 + 90*y^9 + 2945*y^8 + 49960*y^7 + 510160*y^6 + 3377252*y^5 + 14971780*y^4 + 44457000*y^3 + 85336175*y^2 + 96141170*y + 48446971)*x^5/5 + ...

Prog

(PARI) {A360238(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(2*m) *x^m ) +x*O(x^n) ), n, x), n, y)}
{a(n) = polcoeff( exp( sum(m=1, n, A360238(m)*x^m/m ) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A360238, A360349, A000108, A000984.

Sequence in context: A187656 A350376 A255907 * A054260 A053160 A167416

Adjacent sequences: A360236 A360237 A360238 * A360240 A360241 A360242

Keyword

nonn

Author

Paul D. Hanna, Feb 11 2023

Status

approved