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A326276 G.f.: Sum_{n>=0} (1 + (1+x)^(n+1))^n * x^n.

%I A326276 #14 Sep 14 2019 08:46:45
%S A326276 1,2,6,21,85,382,1879,9986,56818,343640,2196596,14770122,104063085,
%T A326276 765661874,5866191429,46683934520,385048724001,3285146877603,
%U A326276 28942067342876,262882422213165,2458316711782337,23637510378534754,233423898596027454,2364847720082290621,24555411743247510317,261085211212909391915
%N A326276 G.f.: Sum_{n>=0} (1 + (1+x)^(n+1))^n * x^n.
%H A326276 Paul D. Hanna, <a href="/A326276/b326276.txt">Table of n, a(n) for n = 0..400</a>
%F A326276 G.f.: Sum_{n>=0} (1 + (1+x)^(n+1))^n * x^n.
%F A326276 G.f.: Sum_{n>=0} (1+x)^(n*(n+1)) * x^n / (1 - x*(1+x)^n)^(n+1).
%e A326276 G.f.: A(x) = 1 + 2*x + 6*x^2 + 21*x^3 + 85*x^4 + 382*x^5 + 1879*x^6 + 9986*x^7 + 56818*x^8 + 343640*x^9 + 2196596*x^10 + ...
%e A326276 such that
%e A326276 A(x) = 1 + (1 + (1+x)^2)*x + (1 + (1+x)^3)^2*x^2 + (1 + (1+x)^4)^3*x^3 + (1 + (1+x)^5)^4*x^4 + ... + (1 + (1+x)^(n+1))^n*x^n + ...
%e A326276 also
%e A326276 A(x) = 1/(1 - x) + (1+x)^2*x/(1 - x*(1+x))^2 + (1+x)^6*x^2/(1 - x*(1+x)^2)^3 + (1+x)^12*x^3/(1 - x*(1+x)^3)^4 + ... + (1+x)^(n*(n+1))*x^n/(1 - x*(1+x)^n)^(n+1) + ...
%e A326276 RELATED SERIES.
%e A326276 Below we illustrate the following identity at specific values of x:
%e A326276 Sum_{n>=0} (1 + (1+x)^(n+1))^n * x^n  =  Sum_{n>=0} (1+x)^(n*(n+1)) * x^n / (1 - x*(1+x)^n)^(n+1).
%e A326276 (1) At x = -1/2, the following sums are equal
%e A326276 S1 = Sum_{n>=0} (-1)^n * (2^(n+1) + 1)^n / 2^(n*(n+2)),
%e A326276 S1 = Sum_{n>=0} (-1)^n * 2 / (2^(n+1) + 1)^(n+1),
%e A326276 where S1 = 0.58938625589631021783349702645576048800172938765646329470992...
%e A326276 (2) At x = -1/3, the following sums are equal
%e A326276 S2 = Sum_{n>=0} (-1)^n * (2^(n+1) + 3^(n+1))^n / 3^(n*(n+2)),
%e A326276 S2 = Sum_{n>=0} (-1)^n * 3 * 2^(n*(n+1)) / (3^(n+1) + 2^n)^(n+1),
%e A326276 where S2 = 0.65707817941052544107009145640756914928885409483935267126701...
%e A326276 (3) At x = -2/3, the following sums are equal
%e A326276 S3 = Sum_{n>=0} (-2)^n * (3^(n+1) + 1)^n / 3^(n*(n+2)),
%e A326276 S3 = Sum_{n>=0} (-2)^n * 3 / (3^(n+1) + 2)^(n+1),
%e A326276 where S3 = 0.55090474258125970373130850821926676214280685554645756713729...
%o A326276 (PARI) {a(n) = polcoeff( sum(m=0,n, (1 + (1+x)^(m+1) +x*O(x^n) )^m * x^m), n)}
%o A326276 for(n=0,30,print1(a(n),", "))
%Y A326276 Cf. A301306.
%K A326276 nonn
%O A326276 0,2
%A A326276 _Paul D. Hanna_, Jun 28 2019