login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A103779 Expansion of real root of y + y^2 + y^3 = x. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

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.

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, sum(j=0..k, C(j,n-k-j) * C(k,j)) * a(k)). - Vladimir Kruchinin, Apr 08 2011

a(n) = 1/n*sum(k=1..n-1, C(k,n-1-k)*(-1)^k*C(n+k-1,n-1)), 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)

EXAMPLE

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.

MATHEMATICA

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

PROG

(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

CROSSREFS

Sequence in context: A244714 A218212 A305637 * A049451 A079776 A117109

Adjacent sequences: A103776 A103777 A103778 * A103780 A103781 A103782

KEYWORD

easy,sign

AUTHOR

Paul Barry, Feb 15 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 3 23:46 EST 2022. Contains 358544 sequences. (Running on oeis4.)