A364374 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^2).
1, 0, -1, 1, 2, -6, -1, 28, -31, -98, 288, 131, -1730, 1638, 7431, -19583, -15502, 135642, -99523, -664050, 1535896, 1816196, -11902728, 5944326, 64487669, -129346490, -213116764, 1112382523, -277762230, -6572175490, 11287106695, 25078981772, -107983368519, -1826241850
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Maple
A364374 := proc(n) add( (-1)^k*binomial(n+k+1,k) * binomial(n+k+1,n-k)/(n+k+1),k=0..n) ; end proc: seq(A364374(n),n=0..80); # R. J. Mathar, Jul 25 2023
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Mathematica
nmax = 33; A[_] = 1; Do[A[x_] = (1+x*A[x])*(1-x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x] (* Jean-François Alcover, Oct 21 2023 *) -
PARI
a(n) = sum(k=0, n, (-1)^k*binomial(n+k+1, k)*binomial(n+k+1, n-k)/(n+k+1));
Formula
a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1).
D-finite with recurrence 15*n*(n+1)*a(n) +2*n*(13*n-11)*a(n-1) +12*(9*n^2-19*n+9)*a(n-2) +2*(10*n^2-65*n+99)*a(n-3) -4*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 25 2023
A(x) = (1/x) * series_reversion(x*(1 + x + x^2)/(1 + x)). - Peter Bala, Sep 08 2024