A264416 - OEIS

%I #9 Nov 18 2015 23:21:26

%S 1,1,3,11,46,234,1269,7609,48501,328831,2347655,17551462,136902440,

%T 1109043889,9308756773,80704240807,721319484280,6632169860334,

%U 62631220229804,606525577239266,6015393685594488,61024457251302195,632560831051022660,6693134988051996106,72226722923184020925,794238562047133369308

%N G.f.: Sum_{n>=0} x^n * (d/dx)^(n^2) x^(n^2) * (1+x)^n / (n^2)!, where (d/dx)^k denotes the k-th derivative operator.

%C Compare to the trivial: Sum_{n>=0} x^n * (d/dx)^(n^2) x^(n^2) / (n^2)! = 1/(1-x).

%H Vaclav Kotesovec, <a href="/A264416/b264416.txt">Table of n, a(n) for n = 0..165</a>

%F  a(n) = Sum_{k=0..[n/2]} binomial(n-k, k) * binomial((n-k)^2 + k, k).

%e G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 234*x^5 + 1269*x^6 + 7609*x^7 + 48501*x^8 + 328831*x^9 + 2347655*x^10 +...

%e where

%e A(x) = 1 + x * d/dx x*(1+x) + x^2 * d^4/dx^4 x^4*(1+x)^2/4! + x^3 * d^9/dx^9 x^9*(1+x)^3/9! + x^4 * d^16/dx^16 x^16*(1+x)^4/16! + x^5 * d^25/dx^25 x^25*(1+x)^5/25! + x^6 * d^36/dx^36 x^36*(1+x)^6/36! +...

%o (PARI) {Dx(n,F)=my(G=F);for(i=1,n,G=deriv(G));G}

%o {a(n) = my(A); A = sum(m=0,n, x^m * Dx(m^2, x^(m^2)*(1+x +x*O(x^n))^m)/(m^2)! ); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial((n-k)^2+k, k))}

%o for(n=0,30,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 17 2015