A369674 -id:A369674 - cp's OEIS frontend

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

Paul D. Hanna, Feb 13 2024

nonn

			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...

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

Cf. A369557, A369674, A369675, A369676.

{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), ", "))

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)).

A370014 a(n) = Product_{k=0..n} (2^k + 4^(n-k)).

Original entry on oeis.org

2, 15, 510, 84240, 69204960, 284844384000, 5892302096179200, 613826012249992396800, 322003239202740297793536000, 850857971372280730568060043264000, 11334246342025651164429104024534384640000, 760681528794595483313206024106936185273712640000

Offset: 0

Paul D. Hanna, Feb 08 2024

nonn

For p > 1, q > 1, limit_{n->oo} ( Product_{k=0..n} (p^k + q^(n-k)) )^(1/n^2) = exp((1/2) * (log(p)^2 + log(p)*log(q) + log(q)^2) / log(p*q)); formula due to Vaclav Kotesovec (cf. A369680). For this sequence, p = 2 and q = 4.

			a(0) = (1 + 1) = 2;
a(1) = (1 + 4)*(2 + 1) = 15;
a(2) = (1 + 4^2)*(2 + 4)*(2^2 + 1) = 510;
a(3) = (1 + 4^3)*(2 + 4^2)*(2^2 + 4)*(2^3 + 1) = 84240;
a(4) = (1 + 4^4)*(2 + 4^3)*(2^2 + 4^2)*(2^3 + 4)*(2^4 + 1) = 69204960;
a(5) = (1 + 4^5)*(2 + 4^4)*(2^2 + 4^3)*(2^3 + 4^2)*(2^4 + 4)*(2^5 + 1) = 284844384000;
...
RELATED SERIES.
Sum_{n>=0} Product_{k=0..n} (1/2^k + 1/4^(n-k)) = 2 + 15/8 + 510/8^3 + 84240/8^6 + 69204960/8^10 + 284844384000/8^15 + 5892302096179200/8^21 + ... + a(n)/8^(n*(n+1)/2) + ... = 5.2656633442570372661094196585300212123165...

Cf. A369673, A369674, A369675, A369676, A369677, A369678, A369679, A369680, A369681.

{a(n) = prod(k=0, n, 2^k + 4^(n-k))}
for(n=0, 15, print1(a(n), ", "))

a(n) = Product_{k=0..n} (2^k + 4^(n-k)).
a(n) = 8^(n*(n+1)/2) * Product_{k=0..n} (1/2^k + 1/4^(n-k)).
a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 2^n/8^k).
a(n) = 2^(n*(n+1)/2) * Product_{k=0..n} (1 + 4^n/8^k).
a(n) = 2^(-n*(n+1)/2) * Product_{k=0..n} (2^n + 8^k).
a(n) = 4^(-n*(n+1)/2) * Product_{k=0..n} (4^n + 8^k).
Limit_{n->oo} a(n)^(1/n^2) = 2^(7/6) = 2.244924096618745962867... [using the formula by Vaclav Kotesovec given in the comments section].

cp's OEIS Frontend

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

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A370014 a(n) = Product_{k=0..n} (2^k + 4^(n-k)).

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cp's OEIS Frontend

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

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Formula

A370014 a(n) = Product_{k=0..n} (2^k + 4^(n-k)).

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PARI

Formula

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