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%I #11 Jul 06 2019 09:38:12
%S 1,4,14,40,155,596,2954,14784,83955,494060,3112246,20505484,141656697,
%T 1020670280,7650120170,59509447736,479350116043,3990968968884,
%U 34287091686110,303500118414180,2764254404207136,25873492982703592,248602031383697602,2449525056696683760,24727324378709830225,255512480002690958696,2700450457984818382502,29169124796849950540212,321787888546754475501470,3623188502069809345093500
%N G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+4).
%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 = 4 and q = (1+x), p = 1, r = x. See other examples for k = 1 (A323680), k = 2 (A326006), k = 3 (A326007).
%F G.f.: Sum_{n>=0} binomial(n+3,n) * x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+4).
%F G.f.: Sum_{n>=0} binomial(n+3,n) * x^n * ((1+x)^n - 1)^n / (1 - x*(1+x)^n)^(n+4).
%F G.f.: Sum_{n>=0} binomial(n+3,n) * x^n * Sum_{k=0..n} binomial(n,k) * ( (1+x)^n - (1+x)^k )^(n-k).
%F G.f.: Sum_{n>=0} binomial(n+3,n) * 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} binomial(n+3,n) * 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} binomial(n-i+3,n-i) * 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} binomial(n-i+3,n-i) * 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 + 4*x + 14*x^2 + 40*x^3 + 155*x^4 + 596*x^5 + 2954*x^6 + 14784*x^7 + 83955*x^8 + 494060*x^9 + 3112246*x^10 + 20505484*x^11 + 141656697*x^12 + ...
%e such that
%e A(x) = 1/(1+x)^4 + 4*x*((1+x) + 1)/(1 + x*(1+x))^5 + 10*x^2*((1+x)^2 + 1)^2/(1 + x*(1+x)^2)^6 + 20*x^3*((1+x)^3 + 1)^3/(1 + x*(1+x)^3)^7 + 35*x^4*((1+x)^4 + 1)^4/(1 + x*(1+x)^4)^8 + 56*x^5*((1+x)^5 + 1)^5/(1 + x*(1+x)^5)^9 + 84*x^6*((1+x)^6 + 1)^6/(1 + x*(1+x)^6)^10 + 120*x^7*((1+x)^7 + 1)^7/(1 + x*(1+x)^7)^11 + ...
%e also,
%e A(x) = 1/(1-x)^4 + 4*x*((1+x) - 1)/(1 - x*(1+x))^5 + 10*x^2*((1+x)^2 - 1)^2/(1 - x*(1+x)^2)^6 + 20*x^3*((1+x)^3 - 1)^3/(1 - x*(1+x)^3)^7 + 35*x^4*((1+x)^4 - 1)^4/(1 - x*(1+x)^4)^8 + 56*x^5*((1+x)^5 - 1)^5/(1 - x*(1+x)^5)^9 + 84*x^6*((1+x)^6 - 1)^6/(1 - x*(1+x)^6)^10 + 120*x^7*((1+x)^7 - 1)^7/(1 - x*(1+x)^7)^11 + ...
%o (PARI) {a(n) = my(A = sum(m=0, n+1, binomial(m+3,m) * x^m*((1+x +x*O(x^n) )^m - 1)^m/(1 - x*(1+x +x*O(x^n) )^m )^(m+4) )); polcoeff(A, n)}
%o for(n=0, 35, print1(a(n), ", "))
%o (PARI) {a(n) = sum(i=0, n, binomial(n-i+3,n-i) * 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, binomial(n-i+3,n-i) * 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, A326006, A326007.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 02 2019
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