OFFSET
0,2
COMMENTS
a(n) == 2 (mod 4) iff n = 2^k for k>=2, and a(n) == 0 (mod 4) elsewhere except at a(0)=1 (conjecture).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
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 * 3^(n-k) * 2^k * x^k].
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 3^(k-j) * 2^j * x^j.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 3^(n-k) * Sum_{j=0..k} C(k,j)^2 * 2^j * x^j.
a(n) = Sum_{k=0..[n/2]} 2^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 3^j.
Recurrence: (n-5)*(n-4)*(n-2)*n^2*a(n) = 4*(n-5)*(n-4)*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - 16*(n-5)*(n-4)*(n-1)*(2*n-3)^2*a(n-2) + 8*(n-5)*(n-2)*(4*n^3 - 28*n^2 + 54*n - 27)*a(n-3) + 24*(n-3)*(5*n^4 - 60*n^3 + 248*n^2 - 408*n + 216)*a(n-4) + 16*(n-4)*(n-1)*(4*n^3 - 44*n^2 + 150*n - 153)*a(n-5) - 64*(n-5)*(n-2)*(n-1)*(2*n-9)^2*a(n-6) + 32*(n-2)*(n-1)*(4*n^3 - 60*n^2 + 298*n - 489)*a(n-7) - 16*(n-6)^2*(n-4)*(n-2)*(n-1)*a(n-8). - Vaclav Kotesovec, Aug 26 2014
a(n) ~ c * d^n / n, where d = 10.094399065494857710014687346... is the root of the equation 16 - 128*d + 256*d^2 - 64*d^3 - 120*d^4 - 32*d^5 + 64*d^6 - 16*d^7 + d^8 = 0, and c = 0.5132545324612697424702223429844481717... . - Vaclav Kotesovec, Aug 26 2014
EXAMPLE
G.f.: A(x) = 1 + 4*x + 24*x^2 + 168*x^3 + 1286*x^4 + 10440*x^5 +...
where the g.f. is given by the binomial series identity:
A(x) = 1/(1-3*x) + x/(1-3*x)^3 * (1 + 2*x) * (1 + 3*x)
+ x^2/(1-3*x)^5 * (1 + 2^2*2*x + 4*x^2) * (1 + 2^2*3*x + 9*x^2)
+ x^3/(1-3*x)^7 * (1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3) * (1 + 3^2*3*x + 3^2*9*x^2 + 27*x^3)
+ x^4/(1-3*x)^9 * (1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*x^4) * (1 + 4^2*3*x + 6^2*9*x^2 + 4^2*27*x^3 + 81*x^4)
+ x^5/(1-3*x)^11 * (1 + 5^2*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*x^5) * (1 + 5^2*3*x + 10^2*9*x^2 + 10^2*27*x^3 + 5^2*81*x^4 + 243*x^5) +...
equals the series
A(x) = 1/(1-x) + x/(1-x)^3 * (1 + x) * (3+2*x)
+ x^2/(1-x)^5 * (1 + 2^2*x + x^2) * (9+2^2*3*2*x+4*x^2)
+ x^3/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (27+3^2*9*2*x+3^2*4*3*x^2+8*x^3)
+ x^4/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*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) * (243+5^2*81*2*x+10^2*27*4*x^2+10^2*9*16*x^3+5^2*3*18*x^4+32*x^5) +...
We can also express the g.f. by another binomial series identity:
A(x) = 1 + x*(3 + (1+2*x)) + x^2*(9 + 2^2*3*(1+2*x) + (1+2^2*2*x+4*x^2))
+ x^3*(27 + 3^2*9*(1+2*x) + 3^2*3*(1+2^2*2*x+4*x^2) + (1+3^2*2*x+3^2*4*x^2+8*x^3))
+ x^4*(81 + 4^2*27*(1+2*x) + 6^2*9*(1+2^2*2*x+4*x^2) + 4^2*3*(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*(243 + 5^2*81*(1+2*x) + 10^2*27*(1+2^2*2*x+4*x^2) + 10^2*9*(1+3^2*2*x+3^2*4*x^2+8*x^3) + 5^2*3*(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)) +...
equals the series
A(x) = 1 + x*(1 + (3+2*x)) + x^2*(1 + 2^2*(3+2*x) + (9+2^2*3*2*x+4*x^2))
+ x^3*(1 + 3^2*(3+2*x) + 3^2*(9+2^2*3*2*x+4*x^2) + (27+3^2*9*2*x+3^2*4*3*x^2+8*x^3))
+ x^4*(1 + 4^2*(3+2*x) + 6^2*(9+2^2*3*2*x+4*x^2) + 4^2*(27+3^2*9*2*x+3^2*4*3*x^2+8*x^3) + (81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*8*x^3+16*x^4))
+ x^5*(1 + 5^2*(3+2*x) + 10^2*(9+2^2*3*2*x+4*x^2) + 10^2*(27+3^2*9*2*x+3^2*4*3*x^2+8*x^3) + 5^2*(81+4^2*27*2*x+6^2*9*4*x^2+4^2*3*8*x^3+16*x^4) + (243+5^2*81*2*x+10^2*27*4*x^2+10^2*9*16*x^3+5^2*3*18*x^4+32*x^5)) +...
PROG
(PARI) /* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-3*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 2^k * x^k) * sum(k=0, m, binomial(m, k)^2 * 3^k * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* By a 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 * 3^(m-k) * 2^k * x^k) * sum(k=0, m, binomial(m, k)^2 * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 3^(m-k) * sum(j=0, k, binomial(k, j)^2 * 2^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * 3^(k-j) * 2^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Formula for a(n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 2^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 3^j))}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 25 2014
STATUS
approved