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A326001 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * ((1+x)^n - 1)^n. 1

%I #8 Jul 02 2019 22:43:13

%S 1,3,24,294,4656,89745,2030628,52649478,1537164243,49869371362,

%T 1778978945148,69186794933664,2912819826915180,131960308762527981,

%U 6400097287173710814,330837181757021507028,18156728772490730152533,1054301496336544498961490,64575610215886231406950048,4160633168635583272785917712,281297401484498036336048099574,19912063300892223213096919150032

%N G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * ((1+x)^n - 1)^n.

%C More generally, the following sums are equal:

%C (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

%C (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),

%C for any fixed integer k; here, k = 3 and p = -1, q = 1+x, r = 1.

%F Generating functions.

%F (1) Sum_{n>=0} (n+1)*(n+2)/2 * ((1+x)^n - 1)^n.

%F (2) Sum_{n>=0} (n+1)*(n+2)/2 * (1+x)^(n^2) / (1 + (1+x)^n)^(n+3).

%e G.f.: A(x) = 1 + 3*x + 24*x^2 + 294*x^3 + 4656*x^4 + 89745*x^5 + 2030628*x^6 + 52649478*x^7 + 1537164243*x^8 + 49869371362*x^9 + 1778978945148*x^10 + ...

%e such that

%e A(x) = 1 + 3*((1+x)-1) + 6*((1+x)^2-1)^2 + 10*((1+x)^3-1)^3 + 15*((1+x)^4-1)^4 + 21*((1+x)^5-1)^5 + 28*((1+x)^6-1)^6 + 36*((1+x)^7-1)^7 + 45*((1+x)^8-1)^8 + 55*((1+x)^9-1)^9 +...

%e is equal to

%e A(x) = 1/2^3 + 3*(1+x)/(1+(1+x))^4 + 6*(1+x)^4/(1+(1+x)^2)^5 + 10*(1+x)^9/(1+(1+x)^3)^6 + 15*(1+x)^16/(1+(1+x)^4)^7 + 21*(1+x)^25/(1+(1+x)^5)^8 + 28*(1+x)^36/(1+(1+x)^6)^9 + 36*(1+x)^49/(1+(1+x)^7)^10 + ...

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

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A122400, A326000.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 01 2019

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