A325214 G.f. A(x) satisfies: 1 = Sum_{n>=0} (1 + x)^(n*(n+1)*(n+2)/3!) / A(x)^n * 1/2^(n+1).

1, 4, 76, 6660, 1214148, 360017028, 155971158524, 92792475869316, 72662261401182820, 72535277610275191076, 89993707128790337826108, 135923590258991068525838404, 245645913626844106827612648116, 523530365452078270295671790495780, 1299582478160711096602245632490049676, 3717486179387360228033345648745010216196, 12140428019954179186668584893988648261704852

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..50

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} (1 + x)^(n*(n+1)*(n+2)/3!) / A(x)^n * 1/2^(n+1).

(2) 1 = Sum_{n>=0} (1 + x)^((n+1)*(n+2)*(n+3)/3!) / A(x)^(n+1) * 1/2^(n+1).

Example

G.f.: A(x) = 1 + 4*x + 76*x^2 + 6660*x^3 + 1214148*x^4 + 360017028*x^5 + 155971158524*x^6 + 92792475869316*x^7 + 72662261401182820*x^8 + ...
such that
1 = 1/2 + (1+x)/(2^2*A(x)) + (1+x)^4/(2^3*A(x)^2) + (1+x)^10/(2^4*A(x)^3) + (1+x)^20/(2^5*A(x)^3) + (1+x)^35/(2^6*A(x)^5) + (1+x)^56/(2^7*A(x)^6) + ...
also,
1 = (1+x)/(2*A(x)) + (1+x)^4/(2*A(x))^2 + (1+x)^10/(2*A(x))^3 + (1+x)^20/(2*A(x))^4 + (1+x)^35/(2*A(x))^5 + (1+x)^56/(2*A(x))^6 + ...

Prog

(PARI) /* Requires suitable precision */
{a(n) = my(A=[1]); for(i=0, n,
A=concat(A, 0); A[#A] = round( polcoeff( sum(n=0, 30*#A+100, (1+x +x*O(x^#A))^(n*(n+1)*(n+2)/3!) / Ser(A)^(n) * 1/2^(n+1)*1.), #A-1)); ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A012010 A012155 A350489 * A118193 A052271 A184272

Adjacent sequences: A325211 A325212 A325213 * A325215 A325216 A325217

Keyword

nonn

Author

Paul D. Hanna, Apr 19 2019

Status

approved