A360239 -id:A360239 - cp's OEIS frontend

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.

2, 42, 1376, 60934, 3377252, 224036904, 17282039280, 1519096411230, 149867251224092, 16398595767212452, 1971137737765484444, 258215735255164847944, 36617351885600586385222, 5588967440618883091216208, 913592455995572681826313856, 159241707066923571547572653630

Offset: 1

Paul D. Hanna, Feb 11 2023

nonn

Related sequence: A000984(n) = binomial(2*n,n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + y)^(2*m) * x^m ) for n >= 1.

			L.g.f.: 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 + 1519096411230*x^8/8 + ...
a(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 + (y^12 + 132*y^11 + 6486*y^10 + 169236*y^9 + 2730921*y^8 + 29547696*y^7 + 224036904*y^6 + 1214958240*y^5 + 4717830978*y^4 + 12868488144*y^3 + 23497266672*y^2 + 25858665696*y + 12994749280)*x^6/6 + ...
Exponentiation yields the g.f. of A360239:
exp(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 + ... + A360239(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A360239, A360348, A266526, A000984, A059304, A098658.

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

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

A360349 G.f. A(x) = exp( Sum_{k>=1} A360348(k) * x^k/k ), where A360348(k) = [y^kx^k/k] log( Sum_{m>=0} (1 + my + y^2)^m * x^m ) for k >= 1.

Original entry on oeis.org

1, 1, 5, 38, 391, 5077, 79535, 1458264, 30621237, 724555611, 19076629520, 553236991215, 17525729241605, 602215048797900, 22312035980459259, 886733059906749795, 37631474149766344476, 1698581174869953607957, 81257725943229600518977, 4106922637708383448243974

Offset: 0

Paul D. Hanna, Feb 12 2023

nonn

Related series: M(x) = exp( Sum_{k>=1} A002426(k) * x^k/k ), where M(x) = 1 + x*M(x) + x^2*M(x)^2 is the Motzkin function (A001006) and A002426(k) = [y^k*x^k/k] log( Sum_{m>=0} (1 + y + y^2)^m * x^m ) for k >= 1.

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 391*x^4 + 5077*x^5 + 79535*x^6 + 1458264*x^7 + 30621237*x^8 + 724555611*x^9 + ...
such that
log(A(x)) = x + 9*x^2/2 + 100*x^3/3 + 1381*x^4/4 + 22771*x^5/5 + 435138*x^6/6 + 9442049*x^7/7 + 229265109*x^8/8 + ... + A360348(n)*x^n/n + ...
where A360348(n) equals the coefficient of y^n*x^n/n in the logarithmic series:
log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) = (y^2 + y + 1)*x + (y^4 + 6*y^3 + 9*y^2 + 6*y + 1)*x^2/2 + (y^6 + 15*y^5 + 63*y^4 + 100*y^3 + 63*y^2 + 15*y + 1)*x^3/3 + (y^8 + 28*y^7 + 242*y^6 + 872*y^5 + 1381*y^4 + 872*y^3 + 242*y^2 + 28*y + 1)*x^4/4 + (y^10 + 45*y^9 + 665*y^8 + 4430*y^7 + 14545*y^6 + 22771*y^5 + 14545*y^4 + 4430*y^3 + 665*y^2 + 45*y + 1)*x^5/5 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A360348, A360239, A186925, A001006, A002426.

{A360348(n) = n * polcoeff( polcoeff( log( sum(m=0, n+1, (1 + m*y + y^2)^m *x^m ) +x*O(x^n) ), n, x), n, y)}
{a(n) = polcoeff( exp( sum(m=1,n, A360348(m)*x^m/m ) +x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

a(n) ~ BesselI(0, 2) * n^n. - Vaclav Kotesovec, Feb 12 2023

cp's OEIS Frontend

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A360349 G.f. A(x) = exp( Sum_{k>=1} A360348(k) * x^k/k ), where A360348(k) = [y^kx^k/k] log( Sum_{m>=0} (1 + my + y^2)^m * x^m ) for k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A360238 a(n) = [y^n*x^n/n] log( Sum_{m>=0} (m + y)^(2*m) * x^m ) for n >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A360349 G.f. A(x) = exp( Sum_{k>=1} A360348(k) * x^k/k ), where A360348(k) = [y^k*x^k/k] log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) for k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A360238 a(n) = [y^nx^n/n] log( Sum_{m>=0} (m + y)^(2m) * x^m ) for n >= 1.

A360349 G.f. A(x) = exp( Sum_{k>=1} A360348(k) * x^k/k ), where A360348(k) = [y^kx^k/k] log( Sum_{m>=0} (1 + my + y^2)^m * x^m ) for k >= 1.