A325578 G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} x^n * (1+x)^(n^2) / A(x)^(n*(n+1)/2).

1, 1, 1, 1, 1, 2, 4, 13, 41, 152, 585, 2408, 10398, 46922, 220594, 1075614, 5429158, 28296558, 152028318, 840671002, 4778156476, 27882129540, 166867893396, 1023291150769, 6424498576966, 41262660744958, 270921037479716, 1817215371471834, 12444490783066300, 86956014104628146, 619630222992962304, 4500360360596423580, 33298628793663193094, 250878144285496830342, 1923797779018220870457, 15008171270194837029808

Offset

0,6

Comments

Compare to: 1+x = Sum_{n>=0} x^n * (1+x)^(n^2) / G(x)^(n*(n+1)/2) holds when G(x) = (1+x)^2.

Links

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

Example

G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 13*x^7 + 41*x^8 + 152*x^9 + 585*x^10 + 2408*x^11 + 10398*x^12 + 46922*x^13 + 220594*x^14 + ...
such that
1/(1-x) = 1 + x*(1+x)/A(x) + x^2*(1+x)^4/A(x)^3 + x^3*(1+x)^9/A(x)^6 + x^4*(1+x)^16/A(x)^10 + x^5*(1+x)^25/A(x)^15 + x^6*(1+x)^36/A(x)^21 + x^7*(1+x)^49/A(x)^28 + x^8*(1+x)^64/A(x)^36 + x^9*(1+x)^81/A(x)^45 + ...

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+x +x*O(x^#A))^(m^2)/Ser(A)^(m*(m+1)/2) - 1)), #A) ); A[n+1]}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Cf. A325579.

Sequence in context: A148255 A148256 A163136 * A118930 A355194 A087214

Adjacent sequences: A325575 A325576 A325577 * A325579 A325580 A325581

Keyword

nonn

Author

Paul D. Hanna, May 22 2019

Status

approved