A179848 Expansion of series reversion of generating function for triangular numbers.

0, 1, -3, 12, -55, 273, -1428, 7752, -43263, 246675, -1430715, 8414640, -50067108, 300830572, -1822766520, 11124755664, -68328754959, 422030545335, -2619631042665, 16332922290300, -102240109897695, 642312451217745, -4048514844039120, 25594403741131680

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..375

Formula

A001764(n) = 0^n - (-1)^n * a(n).

G.f. A(x) satisfies A(x) = x * (1 - A(x))^3.

G.f.: 1 - sinh( arcsinh( sqrt( 27*x/4 ) ) / 3 ) / sqrt( 3*x/4 ).

D-finite with recurrence +2*n*(2*n+1)*a(n) +3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Mar 24 2023

Example

G.f. = x - 3*x^2 + 12*x^3 - 55*x^4 + 273*x^5 - 1428*x^6 + 7752*x^7 - 43263*x^8 + ...

Maple

a:= n-> coeff(series(RootOf(A=x*(1-A)^3, A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2012
# Using function CompInv from A357588.
0, CompInv(23, n -> n*(n+1)/2); # Peter Luschny, Oct 05 2022

Mathematica

CoefficientList[Series[1 - Sinh[ArcSinh[Sqrt[27*x/4]]/3]/Sqrt[3*x/4], {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)

Prog

(PARI) {a(n) = if( n<1, 0, -(-1)^n * (3*n)! / (n! * (2*n+1)!) )};
(PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( x / (1 - x)^3 + x * O(x^n) ), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for( k = 0, n, A = x * (1 - A)^3 ); polcoeff( A, n ))};
(Magma) [n le 0 select 0 else (-1)^(n+1)*Factorial(3*n)/( Factorial(n)* Factorial(2*n+1)): n in [0..30]]; // G. C. Greubel, Aug 14 2018

Crossrefs

Cf. A000217.

Sequence in context: A371429 A361356 A007199 * A001764 A171780 A369159

Adjacent sequences: A179845 A179846 A179847 * A179849 A179850 A179851

Keyword

sign

Author

Michael Somos, Jan 10 2011

Status

approved