A369557 Expansion of Sum_{n>=0} Product_{k=0..n} (x^k + x^(n-k)).

3, 4, 2, 6, 3, 4, 9, 4, 8, 6, 13, 8, 12, 12, 10, 22, 13, 22, 14, 26, 20, 34, 23, 32, 36, 34, 42, 36, 59, 38, 67, 46, 75, 56, 82, 66, 98, 84, 100, 102, 105, 126, 116, 152, 119, 184, 136, 202, 154, 230, 181, 256, 203, 276, 250, 306, 285, 326, 342, 348, 398, 374, 463, 404, 525, 438, 610, 486, 666, 542, 744, 610

Offset: 0

Author

Paul D. Hanna, Feb 06 2024

Keywords

• nonn

Examples

			G.f.: A(x) = 3 + 4*x + 2*x^2 + 6*x^3 + 3*x^4 + 4*x^5 + 9*x^6 + 4*x^7 + 8*x^8 + 6*x^9 + 13*x^10 + 8*x^11 + 12*x^12 + ...
where
A(x) = (1 + 1) + (1 + x)*(x + 1) + (1 + x^2)*(x + x)*(x^2 + 1) + (1 + x^3)*(x + x^2)*(x^2 + x)*(x^3 + 1) + (1 + x^4)*(x + x^3)*(x^2 + x^2)*(x^3 + x)*(x^4 + 1) + (1 + x^5)*(x + x^4)*(x^2 + x^3)*(x^3 + x^2)*(x^4 + x)*(x^5 + 1) + ...
Also,
A(1/x) = (1 + 1) + (1 + x)*(x + 1)/x^2 + (1 + x^2)*(x + x)*(x^2 + 1)/x^6 + (1 + x^3)*(x + x^2)*(x^2 + x)*(x^3 + 1)/x^12 + (1 + x^4)*(x + x^3)*(x^2 + x^2)*(x^3 + x)*(x^4 + 1)/x^20 + (1 + x^5)*(x + x^4)*(x^2 + x^3)*(x^3 + x^2)*(x^4 + x)*(x^5 + 1)/x^30 + ...
For example, at x = 1/2,
A(1/2) = 2 + 9/2^2 + 100/2^6 + 2916/2^12 + 231200/2^20 + 50808384/2^30 + 31258240000/2^42 + 54112148361216/2^56 + 264265663201280000/2^72 + ... + A369673(n)/2^(n*(n+1)) + ... = 6.80013983505192354264...
SPECIFIC VALUES.
A(t) = 4 at t = 0.21135479438007733067820905390237206358880...
A(t) = 5 at t = 0.35111207737762337157349938790010474080253...
A(t) = 6 at t = 0.44509902476179757380223857309576063477813...
A(3/4) = 18.04139246037655138841324835985762487898724341...
A(2/3) = 11.59103511448176661974748662249737201844158309...
A(Phi) = 9.595623356758087506923478384122062088751068609...
A(1/2) = 6.800139835051923542641455169580774467247971025...
A(1/3) = 4.847274134844057155467506697748724715389597193...
A(1/4) = 4.236976626306045459467696438142250301516563681...
A(1/5) = 3.934732308501055907377639201049737298238369356...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2049 from Paul D. Hanna)
• Vaclav Kotesovec, Graph - the asymptotic ratio (400000 terms)

Crossrefs

Cf. A369673, A369674, A369675, A369676, A376542, A376621.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Sum[Product[x^j + x^(k - j), {j, 0, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 08 2024 *)
nmax = 100; CoefficientList[Series[-1 + 2*Sum[x^(k^2) * Product[1 + x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}] + Sum[x^((k-1)*k) * Product[1 + x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 27 2024 *)

PARI

{a(n) = my(A = sum(m=0,n+1, prod(k=0,m, x^k + x^(m-k)) +x*O(x^n) )); polcoeff(A,n)}
for(n=0,70, 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 + x^(n-k)).

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

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

From Vaclav Kotesovec, Sep 29 2024: (Start)

a(n) ~ c * d^sqrt(n) / sqrt(n), where d = A376621 = 2.7510850908889199... and c = sqrt((1 + ((197 - sqrt(27/31)) / 62)^(1/3) + ((197 + sqrt(27/31)) / 62)^(1/3))/3) = 1.146046709280363...

a(n) ~ 4*A376542(n). (End)