A293379 G.f.: exp( Sum_{n>=1} A052886(n) * x^n/n ).

1, 1, 2, 8, 60, 708, 11508, 238356, 5997924, 177615460, 6050014228, 233010132788, 10011985592516, 474815000195844, 24635182609510260, 1388071373846060244, 84406416179170495524, 5509502700599270439588, 384233531128293605883092, 28513258202704586092168820, 2243347226931600749380032388, 186528883284172684793892938308, 16343425532977997699916585695796

Offset

0,3

Comments

The e.g.f. of A052886 equals Series_Reversion( log(1 + x - x^2) ), which also equals (1 - sqrt(5 - 4*exp(x)) )/2.

This sequence is motivated by the following conjectures:

(C1) Given integer series G(x) such that G(0) = G'(0) = 1, define L(n) by

Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(G(x)) )

then exp( Sum_{n>=1} L(n) * x^n/n ) is also an integer series;

(C2) Given G(x) = 1 + x*G(x)^m, define L(n) by

Sum_{n>=1} L(n) * x^n/n! = Series_Reversion( log(G(x)) )

then exp( Sum_{n>=1} L(n) * x^n/n ) = (1 + m*x)/(1 + (m-1)*x).

Links

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

Formula

a(n) ~ sqrt(5) * n^(n-2) / (2^(3/2) * exp(n) * (log(5/4))^(n-1/2)). - Vaclav Kotesovec, Oct 26 2017

a(n) = (1/n)*Sum_{m=1..n} m! * C(m-1) * Sum_{s=m..n} Stirling2(s,m) * a(n-s) for n >= 1 with a(0) = 1, where C(n) = A000108(n). - Petros Hadjicostas, Jun 12 2020

Example

G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 60*x^4 + 708*x^5 + 11508*x^6 + 238356*x^7 + 5997924*x^8 + 177615460*x^9 + 6050014228*x^10 +...
such that
log(A(x)) = x + 3*x^2/2 + 19*x^3/3 + 207*x^4/4 + 3211*x^5/5 + 64383*x^6/6 + 1581259*x^7/7 + 45948927*x^8/8 + 1541641771*x^9/9 +...+ A052886(n)*x^n/n +...
Note that the e.g.f. of A052886 equals Series_Reversion( log(1 + x - x^2) ) = x + 3*x^2/2! + 19*x^3/3! + 207*x^4/4! + 3211*x^5/5! + 64383*x^6/6! + 1581259*x^7/7! + 45948927*x^8/8! +...

Mathematica

nmax = 20; A052886 = CoefficientList[Series[(1 - Sqrt[5 - 4*E^x])/2, {x, 0, nmax}], x] * Range[0, nmax]!; Clear[a]; a[n_]:= a[n] = Sum[A052886[[n-k+1]]*a[k], {k, 0, n-1}] / n; a[0] = 1; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 26 2017 *)

Prog

(PARI) {a(n) = my(A, L); L = x*serlaplace( 1/x*serreverse( log(1+x-x^2 +O(x^(n+2))) ) ); A = exp(L); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) c(n) = (1/(n+1))*binomial(2*n, n); /* A000108 */
lista(nn)={my(a=vector(nn)); a[1]=1; for(n=2, nn, a[n] = (1/(n-1))*sum(m=1, n-1, m!*c(m-1)*sum(s=m, n-1, stirling(s, m, 2)*a[n-s]))); a; } \\ Petros Hadjicostas, Jun 12 2020

Crossrefs

Cf. A000108, A052886, A294115.

Sequence in context: A303532 A355106 A113145 * A294331 A036794 A096121

Adjacent sequences: A293376 A293377 A293378 * A293380 A293381 A293382

Keyword

nonn

Author

Paul D. Hanna, Oct 22 2017

Status

approved