A245929 G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} (-1)^k * C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * (-2*x)^j.

1, -26, 1926, -179780, 18601030, -2040558156, 232474675356, -27194647204296, 3243950895157830, -392816395353590780, 48137032861960009396, -5956254538539775751736, 742934976698338610043676, -93295798612937748051169400, 11783597764983598765508801400

Offset: 0

Author

Paul D. Hanna, Aug 15 2014

Keywords

• sign

Examples

			G.f.: A(x) = 1 - 26*x^4 + 1926*x^8 - 179780*x^12 + 18601030*x^16 -+...
where the g.f. is given by the binomial series:
A(x) = 1 + x*(1 - (1-2*x)) + x^2*(1 - 2^2*(1-2*x) + (1-2^2*2*x+4*x^2))
+ x^3*(1 - 3^2*(1-2*x) + 3^2*(1-2^2*2*x+4*x^2) - (1-3^2*2*x+3^2*4*x^2-8*x^3))
+ x^4*(1 - 4^2*(1-2*x) + 6^2*(1-2^2*2*x+4*x^2) - 4^2*(1-3^2*2*x+3^2*4*x^2-8*x^3) + (1-4^2*2*x+6^2*4*x^2-4^2*8*x^3+16*x^4))
+ x^5*(1 - 5^2*(1-2*x) + 10^2*(1-2^2*2*x+4*x^2) - 10^2*(1-3^2*2*x+3^2*4*x^2-8*x^3) + 5^2*(1-4^2*2*x+6^2*4*x^2-4^2*8*x^3+16*x^4) - (1-5^2*2*x+10^2*4*x^2-10^2*8*x^3+5^2*16*x^4-32*x^5))
+ x^6*(1 - 6^2*(1-2*x) + 15^2*(1-2^2*2*x+4*x^2) - 20^2*(1-3^2*2*x+3^2*4*x^2-8*x^3) + 15^2*(1-4^2*2*x+6^2*4*x^2-4^2*8*x^3+16*x^4) - 6^2*(1-5^2*2*x+10^2*4*x^2-10^2*8*x^3+5^2*16*x^4-32*x^5) + (1-6^2*2*x+15^2*4*x^2-20^2*8*x^3+15^2*16*x^4-6^2*32*x^5+64*x^6)) +...
We can also express the g.f. by the binomial series identity:
A(x) = 1/(1+x) + x/(1+x)^3*(1-x)*(1+2*x)
+ x^2/(1+x)^5*(1 - 2^2*x + x^2)*(1 + 2^2*2*x + 4*x^2)
+ x^3/(1+x)^7*(1 - 3^2*x + 3^2*x^2 - x^3)*(1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3)
+ x^4/(1+x)^9*(1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + x^4)*(1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*x^4)
+ x^5/(1+x)^11*(1 - 5^2*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5)*(1 + 5^2*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*x^5)
+ x^6/(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)*(1 + 6^2*2*x + 15^2*4*x^2 + 20^2*8*x^3 + 15^2*16*x^4 + 6^2*32*x^5 + 64*x^6) +...
Note that all coefficients of x^k in A(x) vanish except for k = 4*n, n>=0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A227845, A245925.

Programs

Mathematica

Table[Sum[Sum[Binomial[4*n-k, k+j]^2 * Binomial[k+j, k]^2 * (-1)^j * 2^k,{j,0,4*n-2*k}],{k,0,2*n}],{n,0,20}] (* Vaclav Kotesovec, Aug 15 2014 *)

PARI

/* By definition: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, (-1)^k*binomial(m, k)^2*sum(j=0, k, binomial(k, j)^2*(-2)^j*x^j)+x*O(x^n))), n)}
for(n=0, 20, print1(a(4*n), ", "))

PARI

/* From binomial identity: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1+x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2*(2*x)^k) * sum(k=0, m, binomial(m, k)^2*(-x)^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 20, print1(a(4*n), ", "))

PARI

/* Formula for a(n), skipping zero-valued terms: */
{a(n)=sum(k=0,n\2,sum(j=k,n-k,binomial(n-k,j)^2*binomial(j,k)^2*(-1)^(k-j)*2^k))}
for(n=0, 20, print1(a(4*n), ", "))

PARI

/* From formula for a(n): */
{a(n)=sum(k=0,2*n,sum(j=0,4*n-2*k,binomial(4*n-k,k+j)^2*binomial(k+j,k)^2*(-1)^j*2^k))}
for(n=0, 20, print1(a(n), ", "))

PARI

/* From formula for a(n): */
{a(n)=sum(k=0, 2*n, binomial(2*k, k)*binomial(2*n+k, 2*n-k)^2*(-2)^(2*n-k))}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} x^n / (1+x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2*(-x)^k] * [Sum_{k=0..n} C(n,k)^2*(2*x)^k].

a(n) = Sum_{k=0..2*n} Sum_{j=0..4*n-2*k} C(4*n-k, k+j)^2 * C(k+j, k)^2 * (-1)^j * 2^k.

a(n) = Sum_{k=0..2*n} C(2*k, k) * C(2*n+k, 2*n-k)^2 * (-2)^(2*n-k).

Recurrence: n^2*(4*n-5)*a(n) = -2*(4*n-3)*(68*n^2 - 102*n + 21)*a(n-1) - 4*(2*n-3)^2 * (4*n-1)*a(n-2). - Vaclav Kotesovec, Aug 15 2014

a(n) ~ sqrt((4+sqrt(18))/16) * (-1)^n * (68+48*sqrt(2))^n / (Pi*n). - Vaclav Kotesovec, Aug 15 2014