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%I #9 Jul 06 2019 09:35:45
%S 1,2,5,10,35,110,484,2090,10449,54526,306394,1817190,11357273,
%T 74488856,510662890,3649746072,27120964497,209070202378,1668632739832,
%U 13763654492458,117143535623438,1027274860573646,9269762507422848,85969437563219710,818533062206441679,7992935182604293880,79974062362514607510,819199175961485536540,8583780760842171037456
%N G.f.: Sum_{n>=0} (n+1) * x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+2).
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n + p)^n / (1 + p*q^n*r)^(n+k),
%C (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * (q^n - p)^n / (1 - p*q^n*r)^(n+k),
%C for any fixed integer k; here, k = 2 and q = (1+x), p = 1, r = x. See other examples for k = 1 (A323680), k = 3 (A326007), k = 4 (A326008).
%F G.f.: Sum_{n>=0} (n+1) * x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+2).
%F G.f.: Sum_{n>=0} (n+1) * x^n * ((1+x)^n - 1)^n / (1 - x*(1+x)^n)^(n+2).
%F G.f.: Sum_{n>=0} (n+1) * x^n * Sum_{k=0..n} binomial(n,k) * ( (1+x)^n - (1+x)^k )^(n-k).
%F G.f.: Sum_{n>=0} (n+1) * x^n * Sum_{k=0..n} binomial(n,k) * ( (1+x)^n + (1+x)^k )^(n-k) * (-1)^k.
%F G.f.: Sum_{n>=0} (n+1) * x^n * Sum_{k=0..n} binomial(n,k) * Sum_{j=0..n-k} (-1)^j * binomial(n-k,j) * (1 + x)^((n-j)*(n-k)).
%F FORMULAS INVOLVING TERMS.
%F a(n) = Sum_{i=0..n} (n-i+1) * Sum_{j=0..n-i} Sum_{k=0..n-i-j} (-1)^k * binomial(n-i,j) * binomial(n-i-j,k) * binomial((n-i-j)*(n-i-k),i).
%F a(n) = Sum_{i=0..n} (n-i+1) * Sum_{j=0..n-i} Sum_{k=0..n-i-j} binomial((n-i-j)*(n-i-k),i) * (-1)^j * (n-i)! / ((n-i-j-k)!*j!*k!).
%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 10*x^3 + 35*x^4 + 110*x^5 + 484*x^6 + 2090*x^7 + 10449*x^8 + 54526*x^9 + 306394*x^10 + 1817190*x^11 + 11357273*x^12 + ...
%e such that
%e A(x) = 1/(1+x)^2 + 2*x*((1+x) + 1)/(1 + x*(1+x))^3 + 3*x^2*((1+x)^2 + 1)^2/(1 + x*(1+x)^2)^4 + 4*x^3*((1+x)^3 + 1)^3/(1 + x*(1+x)^3)^5 + 5*x^4*((1+x)^4 + 1)^4/(1 + x*(1+x)^4)^6 + 6*x^5*((1+x)^5 + 1)^5/(1 + x*(1+x)^5)^7 + 7*x^6*((1+x)^6 + 1)^6/(1 + x*(1+x)^6)^8 + 8*x^7*((1+x)^7 + 1)^7/(1 + x*(1+x)^7)^9 + ...
%e also,
%e A(x) = 1/(1-x)^2 + 2*x*((1+x) - 1)/(1 - x*(1+x))^3 + 3*x^2*((1+x)^2 - 1)^2/(1 - x*(1+x)^2)^4 + 4*x^3*((1+x)^3 - 1)^3/(1 - x*(1+x)^3)^5 + 5*x^4*((1+x)^4 - 1)^4/(1 - x*(1+x)^4)^6 + 6*x^5*((1+x)^5 - 1)^5/(1 - x*(1+x)^5)^7 + 7*x^6*((1+x)^6 - 1)^6/(1 - x*(1+x)^6)^8 + 8*x^7*((1+x)^7 - 1)^7/(1 - x*(1+x)^7)^9 + ...
%o (PARI) {a(n) = my(A = sum(m=0, n+1, (m+1) * x^m*((1+x +x*O(x^n) )^m - 1)^m/(1 - x*(1+x +x*O(x^n) )^m )^(m+2) )); polcoeff(A, n)}
%o for(n=0, 35, print1(a(n), ", "))
%o (PARI) {a(n) = sum(i=0, n, (n-i+1) * sum(j=0, n-i, sum(k=0, n-i-j, (-1)^k * binomial(n-i, j) * binomial(n-i-j, k) * binomial((n-i-j)*(n-i-k), i) )))}
%o for(n=0, 35, print1(a(n), ", "))
%o (PARI) {a(n) = sum(i=0, n, (n-i+1) * sum(j=0, n-i, sum(k=0, n-i-j, (-1)^j * binomial((n-i-j)*(n-i-k), i) * (n-i)! / ((n-i-j-k)!*j!*k!) )))}
%o for(n=0, 35, print1(a(n), ", "))
%Y Cf. A323680, A326007, A326008.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 02 2019
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