A226910 a(n) = Sum_{k=0..floor(n/5)} binomial(n,5*k)*binomial(6*k,k)/(5*k+1).
1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 259, 529, 1189, 3004, 8009, 21073, 53233, 129813, 312733, 763573, 1915251, 4914736, 12720841, 32800186, 83869501, 213261712, 542609237, 1388542312, 3579043987, 9273567337, 24075321925, 62475528190, 161969731985, 419914766965
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..750
Programs
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Mathematica
Table[Sum[Binomial[n,5*k]*Binomial[6*k,k]/(5*k+1),{k,0,Floor[n/5]}],{n,0,20}] (* Vaclav Kotesovec, Jun 28 2013 *) -
PARI
a(n)=sum(k=0,n\5,binomial(n,5*k)*binomial(6*k,k)/(5*k+1)) \\ Charles R Greathouse IV, Jun 24 2013
Formula
Representation in terms of special values of generalized hypergeometric function of type 10F9: a(n) = hypergeom([1/6, 1/3, 1/2, 2/3, 5/6, -(1/5)*n, -(1/5)*n+4/5, -(1/5)*n+3/5, -(1/5)*n+2/5, 1/5-(1/5)*n], [1/5, 2/5, 2/5, 3/5, 3/5, 4/5, 4/5, 1, 6/5], -6^6/5^5), n>=0.
Recurrence: -49781*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*a(n-10) + 10*(n-8)*(n-7)*(n-6)*(n-5)*(26453*n - 123726)*a(n-9) - 15*(n-7)*(n-6)*(n-5)*(40479*n^2 - 351957*n + 782140)*a(n-8) + 120*(n-6)*(n-5)*(7013*n^3 - 87699*n^2 + 378278*n - 565577)*a(n-7) - 6*(n-5)*(148255*n^4 - 2435310*n^3 + 15491085*n^2 - 45173430*n + 50791476)*a(n-6) + 12*(69513*n^5 - 1361100*n^4 + 10838875*n^3 - 43818750*n^2 + 89776250*n - 74437500)*a(n-5) - 93750*(7*n^4 - 98*n^3 + 525*n^2 - 1274*n + 1180)*(n-3)*a(n-4) + 375000*(n-2)*(n^2-6*n+10)*(n-3)^2*a(n-3) - 46875*(n-2)*(n-1)*(3*n^2-15*n+20)*(n-3)*a(n-2) + 31250*(n-2)^2*(n-1)*n*(n-3)*a(n-1) - 3125*(n-2)*(n-1)*n*(n+1)*(n-3)*a(n) = 0. - Vaclav Kotesovec, Jun 28 2013
a(n) ~ (5+6^(1+1/5))^(n+3/2)/(5^(n+1)*6^(1+3/10)*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 28 2013
G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^5 * A(x)^6. - Ilya Gutkovskiy, Jul 25 2021
From Peter Bala, Sep 15 2021: (Start)
O.g.f.: A(x) = (1/x)*series reversion ( x*(1 - x^5)/(1 + x*(1 - x^5)) ).
The g.f. of the m-th binomial transform of this sequence is equal to (1/x)*series reversion ( x*(1 - x^5)/(1 + (m + 1)*x*(1 - x^5)) ). The case m = -1 gives the sequence [1, 0, 0, 0, 0, 1, 0, 0,0, 0, 6, 0, 0, 0, 0, 51, 0, 0, 0, 0, 506, ...] - an aerated version of A002295. (End)