A291536 Expansion of the series reversion of x*(1 + 4*x + x^2)/(1 - x)^4.

1, -8, 101, -1544, 26190, -474144, 8975229, -175492664, 3516970490, -71858843264, 1491301438354, -31349284476496, 666133734882748, -14284509655611840, 308734263333717021, -6718525508918998872, 147085140049822666626, -3237191565662618280384, 71584853778205231503750

Offset: 1

Author

Ilya Gutkovskiy, Aug 25 2017

Keywords

• sign

Comments

Reversion of g.f. for the cubes (A000578).

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Series Reversion
• Index entries for reversions of series

Crossrefs

Cf. A000578, A007297, A263843.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 19}], x], x]]

Formula

G.f. A(x) satisfies: A(x)*(1 + 4*A(x) + A(x)^2)/(1 - A(x))^4 = x.

a(n) ~ -(-1)^n * (5*sqrt(6) - 12) * 2^(3*n-2) * 3^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 11 2017

D-finite with recurrence 7*n*(n-1)*(n+1)*a(n) +4*n*(n-1)*(58*n-83)*a(n-1) -36*(n-1)*(16*n^2-80*n+155)*a(n-2) +432*(-96*n^3+720*n^2-1794*n+1495)*a(n-3) +6912*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Jan 25 2023