A370241 Expansion of Sum_{n>=0} Product_{k=0..n} (x^k*(1+x)^(n-k) + x^(n-k)*(1+x)^k).

3, 6, 15, 36, 98, 258, 677, 1830, 5006, 13340, 35215, 95702, 264851, 717760, 1894473, 5031846, 13788409, 38375030, 105005017, 279236168, 734728565, 1967715202, 5416631023, 15061949148, 41271428388, 110250824636, 289840310574, 766277436248, 2072808806434, 5730605191220

Offset: 0

Author

Paul D. Hanna, Feb 13 2024

Keywords

• nonn

Examples

			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 + ...
where
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) + ...
SPECIFIC VALUES.
A(1/5) = 5.4216712041652671338354486...
A(1/4) = Sum_{n>=0} A369676(n)/4^(n*(n+1)) = 7.1437109433775269577074586...
A(1/3) = Sum_{n>=0} A369675(n)/3^(n*(n+1)) = 19.589361786409617133535937...
A(-1/3) = 1.9743720303058511269360725...
Although the g.f. A(x) diverges at x = -1/2, it may be evaluated formally as
A(-1/2) = Sum_{n>=0} (-1)^n * 2 / 16^(n^2) = 1.875030517549021169...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A369557, A369674, A369675, A369676.

Programs

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)}
for(n=0, 40, print1(a(n), ", "))

Formula

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

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

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

(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)).

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