A118346 Central terms of pendular triangle A118345.

1, 1, 5, 30, 201, 1445, 10900, 85128, 682505, 5585115, 46461437, 391743850, 3340361700, 28755475180, 249572076200, 2181469638880, 19186562661273, 169677521094215, 1507881643936015, 13458730170115778, 120599648894147185

Offset

0,3

Comments

Also, g.f. A(x) = (1/x)*series_reversion of x/(1 + x*GF(A005572)), where GF(A005572) is the g.f. of A005572, which is the number of walks on cubic lattice starting and finishing on the xy plane and never going below it.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: A=A(x) satisfies A = 1 - 2*x*A + 2*x*A^2 + x*A^3.

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

G.f.: A(x) = (1/x)*series_reversion( x*(1-2*x + sqrt((1-2*x)*(1-6*x)))/(2*(1-2*x)) ).

For n>0: a(n) = (1/n)*Sum_{j=0..n} Sum_{i=0..n-1} ( binomial(n,j) * binomial(j,i) * binomial(n-j,2*j-n-i-1) * 5^(2*n-3*j+2*i+1) ). -Vladimir Kruchinin, Dec 26 2010

Mathematica

CoefficientList[1 +InverseSeries[Series[x/((1+x)*(1+4*x+x^2)), {x, 0, 30}]], x] (* G. C. Greubel, Mar 17 2021 *)

Prog

(PARI) {a(n) = polcoeff(serreverse( x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^n)))/(2*(1-2*x)))/x, n)}
(Sage)
def A118346_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( ( x/((1+x)*(1+4*x+x^2)) ).reverse() ).list()
a=A118346_list(31); [1]+a[1:] # G. C. Greubel, Mar 17 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
[1] cat Coefficients(R!( Reversion( x/((1+x)*(1+4*x+x^2)) ) )); // G. C. Greubel, Mar 17 2021

Crossrefs

Cf. A118345, A118347, A118348, A118349.

Sequence in context: A196678 A128328 A245376 * A234422 A091927 A253076

Adjacent sequences: A118343 A118344 A118345 * A118347 A118348 A118349

Keyword

nonn

Author

Paul D. Hanna, Apr 26 2006

Status

approved