A325453 G.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n^2) = Sum_{n>=0} x^n / [Product_{k=1..n} 1 - (2*k-1)*x].

1, 1, 1, 4, 22, 139, 1030, 8445, 74766, 705916, 7052531, 74161365, 817741825, 9428968826, 113449665799, 1421957940220, 18538429347923, 251062976257349, 3527558671120766, 51360468675879418, 774005747590247002, 12059539535205433827, 194049771204345008848, 3221298357944442093597, 55111256702479720092044, 970767762280740604708491, 17589320907639742684402262, 327532084150359052415186316

Offset

0,4

Links

Table of n, a(n) for n=0..27.

Example

G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 22*x^4 + 139*x^5 + 1030*x^6 + 8445*x^7 + 74766*x^8 + 705916*x^9 + 7052531*x^10 + 74161365*x^11 + 817741825*x^12 + ...
such that
B(x) = 1 + x*A(x) + x^2*A(x)^4 + x^3*A(x)^9 + x^4*A(x)^16 + x^5*A(x)^25 + x^6*A(x)^36 + x^7*A(x)^49 + x^8*A(x)^64 + x^9*A(x)^81 + ...
B(x) = 1 + x/(1-x) + x^2/((1-x)*(1-3*x)) + x^3/((1-x)*(1-3*x)*(1-5*x)) + x^4/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)) + x^5/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)*(1-9*x)) + x^6/((1-x)*(1-3*x)*(1-5*x)*(1-7*x)*(1-9*x)*(1-11*x)) + ....
where
B(x) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 116*x^5 + 648*x^6 + 4088*x^7 + 28640*x^8 + 219920*x^9 + 1832224*x^10 + 16430176*x^11 + 157554048*x^12 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, x^m*( 1/prod(k=1, m, 1-(2*k-1)*x +x*O(x^#A)) - Ser(A)^(m^2)) ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A325452.

Sequence in context: A200731 A193116 A187254 * A216712 A240586 A369156

Adjacent sequences: A325450 A325451 A325452 * A325454 A325455 A325456

Keyword

nonn

Author

Paul D. Hanna, Apr 30 2019

Status

approved