A305107 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x)^2 ) * (n + 1 - A(x)) = 0 for n > 0.

1, 1, 10, 216, 7852, 427770, 32649276, 3333409849, 439648389640, 72863444853189, 14835946021507520, 3642615447410904525, 1061681881255681884336, 362470236144441144939674, 143310411318629778406908494, 64968204494588611586367685020, 33478892881907679134025607700400, 19460912067689653469231875029090451, 12674293598137775224869798728198782626, 9192057681791476282831341020711249418814

Offset

0,3

Comments

Note: given F(x) = 1 + x * d/dx x*F(x)^2, where x*F(x) is a g.f. of A000699, then

(1) [x^n] exp( x*F(x)^2 ) * (n + 1 - F(x)) = 0 for n > 0,

(2) [x^n] exp( n * x*F(x)^2 ) * (2 - F(x)) = 0 for n > 0.

It is remarkable that this sequence should consist entirely of integers.

Links

Table of n, a(n) for n=0..19.

Example

O.g.f.: A(x) = 1 + x + 10*x^2 + 216*x^3 + 7852*x^4 + 427770*x^5 + 32649276*x^6 + 3333409849*x^7 + 439648389640*x^8 + 72863444853189*x^9 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 21*x^2 + 452*x^3 + 16236*x^4 + 875564*x^5 + 66357788*x^6 + 6744065714*x^7 + 886863035042*x^8 + 146693676869950*x^9 + ...
exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 139*x^3/3! + 11425*x^4/4! + 2009141*x^5/5! + 643102861*x^6/6! + 339114884935*x^7/7! + 274704279360449*x^8/8! + ...

Prog

(PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)^2) ) * (m - Ser(A)) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A305108.

Sequence in context: A274763 A002967 A243476 * A374794 A294850 A259189

Adjacent sequences: A305104 A305105 A305106 * A305108 A305109 A305110

Keyword

nonn

Author

Paul D. Hanna, May 27 2018

Status

approved