cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A360348 a(n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + m*y + y^2)^m * x^m ) for n >= 1.

Original entry on oeis.org

1, 9, 100, 1381, 22771, 435138, 9442049, 229265109, 6160375990, 181559237499, 5826147967201, 202295647539886, 7559401377952659, 302570522540568557, 12917629672442043340, 586047019821392518293, 28159186576616423049683, 1428679795354280280080736, 76329278834398327082152543
Offset: 1

Views

Author

Paul D. Hanna, Feb 12 2023

Keywords

Comments

Related sequence: A002426(n) = [y^n*x^n/n] log( Sum_{m>=0} (1 + y + y^2)^m * x^m ) for n >= 1.

Examples

			L.g.f.: 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 + 6160375990*x^9/9 + ...
a(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 + (y^12 + 66*y^11 + 1491*y^10 + 16002*y^9 + 91293*y^8 + 281220*y^7 + 435138*y^6 + 281220*y^5 + 91293*y^4 + 16002*y^3 + 1491*y^2 + 66*y + 1)*x^6/6 + ...
Exponentiation yields the g.f. of A360349:
exp(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 + ... + A360349(n)*x^n + ...

Links

Crossrefs

Cf. A360349, A360238, A002426.

Programs

  • PARI
    {a(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)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ BesselI(0, 2) * n^(n+1). - Vaclav Kotesovec, Feb 12 2023

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.

Original entry on oeis.org

1, 2, 23, 502, 16414, 716936, 39167817, 2567058766, 196159319943, 17118727499178, 1679643875717867, 183020512751712554, 21928106267349661127, 2865208654370111795940, 405479888251812823615679, 61785441098476295018209264, 10085622916281496742096639996
Offset: 0

Views

Author

Paul D. Hanna, Feb 11 2023

Keywords

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.

Examples

			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 + ...

Links

Crossrefs

Cf. A360238, A360349, A000108, A000984.

Programs

  • 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),", "))

Formula

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

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