A104632 1/n times A104631(n), the coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.
1, 2, 6, 20, 73, 281, 1125, 4635, 19525, 83710, 364070, 1602327, 7123041, 31937010, 144255802, 655804649, 2998354717, 13777825186, 63596593430, 294743653360, 1371017707245, 6398580086645, 29952930770185, 140604572777250, 661708404611603, 3121439743413256, 14756658303857332
Offset: 1
Links
- C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.
Crossrefs
Cf. A005717 (coefficient of x^(n+1) in the expansion of (1+x+x^2)^n).
Programs
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Mathematica
f=1; Table[f=Expand[f(x^4+x^3+x^2+x+1)]; Coefficient[f, x, 2n+1]/n, {n, 30}] a[ n_] := If[ n < 1, 0, Coefficient[ (1 + x + x^2 + x^3 + x^4)^n, x, 2 n + 1] / n]; (* Michael Somos, Dec 01 2016 *) -
Maxima
a(n):=sum((-1)^i*binomial(n,i)*binomial(3*n-5*i,n-1),i,0,(2*n+1)/5)/n; /* Vladimir Kruchinin, Apr 06 2017 */
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PARI
a(n) = polcoeff((1+x+x^2+x^3+x^4)^n, 2*n+1)/n \\ Michel Marcus, Sep 24 2016
Formula
a(n) = Sum_{i=0..(2*n+1)/5}((-1)^i*binomial(n,i)*binomial(3*n-5*i,n-1))/n. - Vladimir Kruchinin, Apr 06 2017
Conjecture: 2*n*(2*n+1)*(n-1)*a(n) -(n-1)*(19*n^2-19*n+2)*a(n-1) -5*(n-2)*(2*n^2-3*n-1)*a(n-2) +25*n*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 23 2017
Comments