A306463 a(n) = Sum_{k=0..n} Sum_{m=0..floor(k/2)} binomial(k-m, m)*binomial(n-k, k-m)^2.
1, 1, 2, 6, 15, 37, 98, 262, 699, 1883, 5110, 13918, 38045, 104355, 287028, 791320, 2186209, 6051113, 16776022, 46577806, 129491865, 360432855, 1004332322, 2801307498, 7820572153, 21851390549, 61101872126, 170977916730, 478755116117, 1341389394715, 3760507521800
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..2202
Programs
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Maxima
a(n):=sum(sum(binomial(k-m,m)*binomial(n-k,k-m)^2,m,0,k/2),k,0,n);
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PARI
a(n) = sum(k=0, n, sum(m=0, k\2, binomial(k-m, m)*binomial(n-k, k-m)^2)); \\ Michel Marcus, Feb 18 2019
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PARI
N=66; x='x+O('x^N); Vec(1/sqrt(x^6+2*x^5-x^4-4*x^3-x^2-2*x+1)) \\ Seiichi Manyama, Feb 20 2019
Formula
G.f.: 1/sqrt(x^6 + 2*x^5 - x^4 - 4*x^3 - x^2 - 2*x + 1).
D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3) +(-n+2)*a(n-4) +(2*n-5)*a(n-5) +(n-3)*a(n-6)=0. - R. J. Mathar, Jan 16 2020