A326000 - OEIS

%I #10 Jul 02 2019 22:42:13

%S 1,2,12,120,1607,26862,536816,12466468,329648274,9774030812,

%T 321057111308,11570735358300,453874209520951,19248243764760562,

%U 877497573254643438,42791783608096161848,2222646606788322292656,122500263059540271947448,7140154262067048381368062,438819217371889984410077532,28360033818941846664929891481,1922734355204851243123303962324

%N G.f.: Sum_{n>=0} (n+1) * ((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 = 2 and p = -1, q = 1+x, r = 1.

%F Generating functions.

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

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

%e G.f.: A(x) = 1 + 2*x + 12*x^2 + 120*x^3 + 1607*x^4 + 26862*x^5 + 536816*x^6 + 12466468*x^7 + 329648274*x^8 + 9774030812*x^9 + 321057111308*x^10 + ...

%e such that

%e A(x) = 1 + 2*((1+x)-1) + 3*((1+x)^2-1)^2 + 4*((1+x)^3-1)^3 + 5*((1+x)^4-1)^4 + 6*((1+x)^5-1)^5  + 7*((1+x)^6-1)^6 + 8*((1+x)^7-1)^7 + 9*((1+x)^8-1)^8 + 10*((1+x)^9-1)^9 +...

%e is equal to

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

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

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

%Y Cf. A122400, A326001.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 01 2019