A246570 G.f.: Sum_{n>=0} x^n / (1-x)^(4*n+1) * [Sum_{k=0..2*n} C(2*n,k)^2 * x^k]^2.
1, 2, 15, 116, 1001, 9322, 89363, 881376, 8860677, 90407666, 933482527, 9731366060, 102259648701, 1081810639970, 11510355762339, 123077391281248, 1321739147949829, 14248409211657754, 154118033900091139, 1672053762899099700, 18189628173538580233, 198362957005290443978
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 15*x^2 + 116*x^3 + 1001*x^4 + 9322*x^5 + 89363*x^6 +... where A(x) = 1/(1-x) + x/(1-x)^5 * (1 + 2^2*x + x^2)^2 + x^2/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2 + x^3/(1-x)^13 * (1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^2 +... The square-root of the g.f. is an integer series: A(x)^(1/2) = 1 + x + 7*x^2 + 51*x^3 + 425*x^4 + 3879*x^5 + 36527*x^6 + 355333*x^7 + 3531175*x^8 + 35673875*x^9 +...+ A246572(n)*x^n +...
Links
- Vaclav Kotesovec, Recurrence (of order 8)
Programs
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Mathematica
Table[Sum[Sum[Binomial[2*n-k-j,k]^2 * Binomial[k,j]^2,{j,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 02 2014 *) -
PARI
/* By definition: */ {a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(4*m+1) * sum(k=0, 2*m, binomial(2*m, k)^2 * x^k)^2 +x*O(x^n)); polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) -
PARI
/* From a formula for a(n): */ {a(n)=sum(k=0, n, sum(j=0, min(k, 2*n-2*k), binomial(2*n-k-j, k)^2 * binomial(k, j)^2 ))} for(n=0, 25, print1(a(n), ", "))
Formula
a(n) = Sum_{k=0..n} Sum_{j=0..k} C(2*n-k-j,k)^2 * C(k,j)^2.
Comments