A259550 a(n) = C(5*n-1,2*n)/3, n > 0, a(0) = 1.
1, 2, 42, 1001, 25194, 653752, 17298645, 463991880, 12570420330, 343176898988, 9425842448792, 260170725132045, 7210477496434485, 200519284375732896, 5592628786362932776, 156375886125188595376, 4382048530314336892010, 123033460966787345446836
Offset: 0
Links
- D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
- V. V. Kruchinin and D. V. Kruchinin, Composita and its properties, J. Analysis and Number Theory 2 (2014), 1-8.
- V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
- D. V. Kruchinin, On solving some functional equations, Advances in Difference Equations, Vol. 1 (2015), 1687-1847.
Programs
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Magma
[1] cat [Binomial(5*n-1, 2*n)/3: n in [1..20]]; // Vincenzo Librandi, Jul 01 2015
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Mathematica
Join[{1}, Table[Binomial[5 n - 1, 2 n]/3, {n, 30}]] (* Vincenzo Librandi, Jul 01 2015 *) -
Maxima
makelist(if n=0 then 1 else binomial(5*n-1,2*n)/3,n,0,20);
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PARI
vector(20, n, n--; if (n==0, 1, binomial(5*n-1,2*n)/3)) \\ Michel Marcus, Jul 01 2015
Formula
G.f.: A(x) = 1 + (x*B(x)')/(B(x)), B(x) = (1 + x*B(x)^5)*C(x*B(x)^5) is g.f. of A060941, C(x) is g.f. of Catalan numbers.
a(n) = n*Sum_{i = 0..n}((C(5*n,i)*C(7*n-2*i-1,n-i))/(6*n-i)), n > 1, a(0) = 1.
a(n) = 1/5*A001450(n) for n >= 1. exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 23*x^2 + 377*x^3 + ... is the o.g.f. for the sequence of Duchon numbers A060941. - Peter Bala, Oct 05 2015
D-finite with recurrence 6*n*(3*n-1)*(2*n-1)*(3*n-2)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024