OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f.: Sum_{n>=0} x^n * (1 + (1+x)^n)^n / (1 - x*(1+x)^n)^(n+1).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * ( (1+x)^n + (1+x)^k )^(n-k).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * (1 + x)^((n-j)*(n-k)).
FORMULAS INVOLVING TERMS.
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} binomial(n-i,j) * binomial(n-i-j,k) * binomial((n-i-j)*(n-i-k),i).
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} binomial((n-i-j)*(n-i-k),i) * (n-i)! / ((n-i-j-k)!*j!*k!).
EXAMPLE
G.f.: A(x) = 1 + 3*x + 10*x^2 + 41*x^3 + 190*x^4 + 973*x^5 + 5413*x^6 + 32351*x^7 + 205966*x^8 + 1387807*x^9 + 9845083*x^10 + 73215780*x^11 + 568757151*x^12 + ...
such that
A(x) = 1/(1-x) + x*(1 + (1+x))/(1 - x*(1+x))^2 + x^2*(1 + (1+x)^2)^2/(1 - x*(1+x)^2)^3 + x^3*(1 + (1+x)^3)^3/(1 - x*(1+x)^3)^4 + x^4*(1 + (1+x)^4)^4/(1 - x*(1+x)^4)^5 + x^5*(1 + (1+x)^5)^5/(1 - x*(1+x)^5)^6 + x^6*(1 + (1+x)^6)^6/(1 - x*(1+x)^6)^7 + x^7*(1 + (1+x)^7)^7/(1 - x*(1+x)^7)^8 + ...
PROG
(PARI) {a(n) = my(A = sum(m=0, n+1, x^m*((1+x +x*O(x^n) )^m + 1)^m/(1 - x*(1+x +x*O(x^n) )^m )^(m+1) )); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = sum(i=0, n, sum(k=0, n-i, sum(j=0, n-i-k, binomial(n-i, k) * binomial(n-i-k, j) * binomial((n-i-k)*(n-i-j), i) )))}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, binomial((n-i-j)*(n-i-k), i) * (n-i)! / ((n-i-j-k)!*j!*k!) )))}
for(n=0, 35, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2019
STATUS
approved