A103779 Series reversion of y + y^2 + y^3.

0, 1, -1, 1, 0, -4, 14, -30, 33, 55, -429, 1365, -2652, 1428, 12920, -64600, 178296, -277932, -152950, 2770350, -10785390, 25312650, -26053020, -84847620, 576753450, -1856900682, 3566658438, -843350102, -24973594296, 117328602840, -317641049880, 455822225496

Offset: 0

Author

Paul Barry, Feb 15 2005

Keywords

• easy
• sign

Comments

Second column of A103778 (inverse of trinomial triangle A071675).

Examples

			G.f.: A(x) = x - x^2 + x^3 - 4*x^5 + 14*x^6 - 30*x^7 + 33*x^8 + 55*x^9 - 429*x^10 + 1365*x^11 - 2652*x^12 + 1428*x^13 + 12920*x^14 + ... where A(x + x^2 + x^3) = x.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 0..103
• Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Crossrefs

Cf. A350383.

Programs

Mathematica

CoefficientList[ InverseSeries[ Series[y + y^2 + y^3, {y, 0, 28}], x], x] (* Robert G. Wilson v *)

Maxima

a(n):=if n=1 then 1 else -sum(sum(binomial(j,n-k-j) *binomial(k,j),j,0,k)*a(k),k,1,n-1); /* Vladimir Kruchinin, Apr 08 2011 */

Maxima

a(n):=if n=1 then 1 else 1/n*sum(binomial(k,n-1-k)*(-1)^k *binomial(n+k-1,n-1),k,1,n-1); /* Vladimir Kruchinin, May 12 2012 */

PARI

Vec(serreverse(x*(1+x+x^2)+O(x^66))) /* Joerg Arndt, Aug 19 2012 */

PARI

/* G.f. A(x) satisfies: A(x)^2 = A( x^2 - 2*x*A(x)^2 ) */
{a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^2 - 2*X*A^2)^(1/2) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Apr 17 2016

Formula

G.f.: -2^(2/3) * ((3*sqrt(3)*sqrt(27*x^2+14*x+3)-27*x-7)^(1/3) -(3*sqrt(3) * sqrt(27*x^2+14*x+3)+27*x+7)^(1/3) +2^(1/3))/6.

a(0) = 0, a(1) = 1, and, for n >= 2, a(n) = -Sum_{k = 1..n-1} a(k) * Sum_{j = 0..k} C(j,n-k-j) * C(k,j). - Vladimir Kruchinin, Apr 08 2011

a(n) = (1/n)*Sum_{k = 1..n-1} (-1)^k * C(k,n-1-k) * C(n+k-1,n-1), with a(1) = 1. - Vladimir Kruchinin, May 12 2012

D-finite with recurrence 3*n*(n-1)*a(n) +7*(n-1)*(2*n-3)*a(n-1) +3*(3*n-5)*(3*n-7)*a(n-2)=0. - R. J. Mathar, Oct 06 2012

G.f. A(x) satisfies: A(x)^2 = A( x^2 - 2*x*A(x)^2 ). - Paul D. Hanna, Apr 17 2016

From Paul D. Hanna, Sep 06 2022: (Start)

G.f. A(x) satisfies:

A(x)^5 = A( x^5 - 5*x*(1+x)^2*A(x)^5 ), and

A(x)^5 = ( x^5 - 5*x*(1+x)^2*A(x)^5 ) * (1 - A(x)^5) / (1 - A(x)^15). (End)

A(x) = x*exp(Sum_{n >= 1} A350383(n)*x^n/n). - Peter Bala, Mar 11 2025