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%I #13 Jul 06 2019 09:37:06
%S 1,3,9,22,81,285,1339,6264,33567,186811,1116105,7001244,46150265,
%T 318158826,2286494076,17088720336,132492477111,1063527470481,
%U 8822541504319,75512660179788,665878308902676,6041491458457319,56330651731617333,539160888285121116,5292067580412471801,53218232521845617886,547833354998854396224,5768212264434778469998,62074688689939991197548
%N G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+3).
%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 = 3 and q = (1+x), p = 1, r = x. See other examples for k = 1 (A323680), k = 2 (A326006), k = 4 (A326008).
%F G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * ((1+x)^n + 1)^n / (1 + x*(1+x)^n)^(n+3).
%F G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^n * ((1+x)^n - 1)^n / (1 - x*(1+x)^n)^(n+3).
%F G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * 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)*(n+2)/2 * 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)*(n+2)/2 * 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)*(n-i+2)/2 * 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)*(n-i+2)/2 * 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 + 3*x + 9*x^2 + 22*x^3 + 81*x^4 + 285*x^5 + 1339*x^6 + 6264*x^7 + 33567*x^8 + 186811*x^9 + 1116105*x^10 + 7001244*x^11 + 46150265*x^12 + ...
%e such that
%e A(x) = 1/(1+x)^3 + 3*x*((1+x) + 1)/(1 + x*(1+x))^4 + 6*x^2*((1+x)^2 + 1)^2/(1 + x*(1+x)^2)^5 + 10*x^3*((1+x)^3 + 1)^3/(1 + x*(1+x)^3)^6 + 15*x^4*((1+x)^4 + 1)^4/(1 + x*(1+x)^4)^7 + 21*x^5*((1+x)^5 + 1)^5/(1 + x*(1+x)^5)^8 + 28*x^6*((1+x)^6 + 1)^6/(1 + x*(1+x)^6)^9 + 36*x^7*((1+x)^7 + 1)^7/(1 + x*(1+x)^7)^10 + ...
%e also,
%e A(x) = 1/(1-x)^3 + 3*x*((1+x) - 1)/(1 - x*(1+x))^4 + 6*x^2*((1+x)^2 - 1)^2/(1 - x*(1+x)^2)^5 + 10*x^3*((1+x)^3 - 1)^3/(1 - x*(1+x)^3)^6 + 15*x^4*((1+x)^4 - 1)^4/(1 - x*(1+x)^4)^7 + 21*x^5*((1+x)^5 - 1)^5/(1 - x*(1+x)^5)^8 + 28*x^6*((1+x)^6 - 1)^6/(1 - x*(1+x)^6)^9 + 36*x^7*((1+x)^7 - 1)^7/(1 - x*(1+x)^7)^10 + ...
%o (PARI) {a(n) = my(A = sum(m=0, n+1, (m+1)*(m+2)/2 * x^m*((1+x +x*O(x^n) )^m - 1)^m/(1 - x*(1+x +x*O(x^n) )^m )^(m+3) )); polcoeff(A, n)}
%o for(n=0, 35, print1(a(n), ", "))
%o (PARI) {a(n) = sum(i=0, n, (n-i+1)*(n-i+2)/2 * 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)*(n-i+2)/2 * 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, A326008.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 02 2019
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