A305602 G.f. A(x) satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^2)].
1, 1, 6, 54, 628, 8760, 140904, 2552151, 51243864, 1127982321, 26993774100, 697703846499, 19372450060296, 575205186725962, 18191422973198622, 610655961723782310, 21689599103526363600, 812832263931582168447, 32057155649057309677062, 1327393477257351399000744, 57581802198755959140129600
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 628*x^4 + 8760*x^5 + 140904*x^6 + 2552151*x^7 + 51243864*x^8 + 1127982321*x^9 + 26993774100*x^10 + ... such that A(x) = 1 + x*[d/dx 1/(1 - x*A(x)^2)]. RELATED SERIES. A(x)^2 = 1 + 2*x + 13*x^2 + 120*x^3 + 1400*x^4 + 19424*x^5 + 309780*x^6 + 5559054*x^7 + 110623342*x^8 + 2415298374*x^9 + 57387784542*x^10 + ... 1/(1 - x*A(x)^2) = 1 + x + 3*x^2 + 18*x^3 + 157*x^4 + 1752*x^5 + 23484*x^6 + 364593*x^7 + 6405483*x^8 + 125331369*x^9 + 2699377410*x^10 + ... exp( Integral A(x)^2 dx ) = 1 + x + 3*x^2/2! + 33*x^3/3! + 849*x^4/4! + 38061*x^5/5! + 2575611*x^6/6! + 242377533*x^7/7! + 30085188993*x^8/8! + ... A'(x)/A(x) = 1 + 11*x + 145*x^2 + 2247*x^3 + 39461*x^4 + 768983*x^5 + 16409646*x^6 + 380013063*x^7 + 9487631035*x^8 + 254076973011*x^9 + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Programs
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PARI
{a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A^2+x*O(x^n)))); polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) -
PARI
{a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*intformal(Ser(A)^2) ) * ((m-1) + 1 - Ser(A)) )[m] ); A[n+1]} for(n=0, 25, print1(a(n), ", "))
Formula
O.g.f. A(x) satisfies:
(1) [x^n] exp( n * Integral A(x)^2 dx ) * (n + 1 - A(x)) = 0 for n > 0.
(2) A(x) = 1 + x*A(x)*(A(x) + 2*x*A'(x))/(1 - x*A(x)^2)^2.
a(n) ~ c * 2^n * n^(3/2) * n!, where c = 0.26934871195193907483980578... - Vaclav Kotesovec, Oct 06 2020