%I #6 Feb 13 2024 05:16:14

%S 3,6,15,36,98,258,677,1830,5006,13340,35215,95702,264851,717760,

%T 1894473,5031846,13788409,38375030,105005017,279236168,734728565,

%U 1967715202,5416631023,15061949148,41271428388,110250824636,289840310574,766277436248,2072808806434,5730605191220

%N Expansion of Sum_{n>=0} Product_{k=0..n} (x^k*(1+x)^(n-k) + x^(n-k)*(1+x)^k).

%H Paul D. Hanna, <a href="/A370241/b370241.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

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

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

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

%F (4) A(x/(1-x)) = Sum_{n>=0} 1/(1-x)^(n*(n+1)) * Product_{k=0..n} (x^k + x^(n-k)).

%e G.f.: A(x) = 3 + 6*x + 15*x^2 + 36*x^3 + 98*x^4 + 258*x^5 + 677*x^6 + 1830*x^7 + 5006*x^8 + 13340*x^9 + 35215*x^10 + 95702*x^11 + 264851*x^12 + ...

%e where

%e A(x) = (1 + 1) + ((1+x) + x)*(x + (1+x)) + ((1+x)^2 + x^2)*(x*(1+x) + x*(1+x))*(x^2 + (1+x)^2) + ((1+x)^3 + x^3)*(x*(1+x)^2 + x^2*(1+x))*(x^2*(1+x) + x*(1+x)^2)*(x^3 + (1+x)^3) + ((1+x)^4 + x^4)*(x*(1+x)^3 + x^3*(1+x))*(x^2*(1+x)^2 + x^2*(1+x)^2)*(x^3*(1+x) + x*(1+x)^3)*(x^4 + (1+x)^4) + ...

%e SPECIFIC VALUES.

%e A(1/5) = 5.4216712041652671338354486...

%e A(1/4) = Sum_{n>=0} A369676(n)/4^(n*(n+1)) = 7.1437109433775269577074586...

%e A(1/3) = Sum_{n>=0} A369675(n)/3^(n*(n+1)) = 19.589361786409617133535937...

%e A(-1/3) = 1.9743720303058511269360725...

%e Although the g.f. A(x) diverges at x = -1/2, it may be evaluated formally as

%e A(-1/2) = Sum_{n>=0} (-1)^n * 2 / 16^(n^2) = 1.875030517549021169...

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

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A369557, A369674, A369675, A369676.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Feb 13 2024