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A118346
Central terms of pendular triangle A118345.
6
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 26 2006
STATUS
approved