A386565 a(n) = Sum_{k=0..n-1} binomial(4*k-1,k) * binomial(4*n-4*k,n-k-1).
0, 1, 11, 111, 1091, 10596, 102237, 982458, 9415539, 90063180, 860278156, 8208539351, 78258171957, 745595635084, 7099714918062, 67574576298276, 642927956583123, 6115089154367484, 58146652079312580, 552769690436583532, 5253812277363417836, 49925987913040522128
Offset: 0
Keywords
Examples
(1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ) = x + 11*x^2/2 + 37*x^3 + 1091*x^4/4 + 10596*x^5/5 + ...
Programs
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PARI
a(n) = sum(k=0, n-1, binomial(4*k-1, k)*binomial(4*n-4*k, n-k-1));
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PARI
my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(4*k, k)/(3*k+1)*x^k)); concat(0, Vec(g*(g-1)/(4-3*g)^2))
Formula
G.f.: g*(g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/(1-4*g)^2 where g*(1-g)^3 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(4*k-1+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k,k).