0,3

Conjecture: a(n) is the number of partitions p of n into distinct parts such that max(p) <= 1 + 2*min(p), for n >= 1 (as in the Mathematica program at A241061). - Clark Kimberling, Apr 16 2014

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)

G.f.: Sum_{n>=0} x^(n*(3*n+1)/2) / ( Product_{k=0..n} 1 - x^(n+k+1) ). - Paul D. Hanna, Oct 14 2020

a(n) ~ c * exp(r*sqrt(n)) / sqrt(n), where r = 0.926140105877... = 2*sqrt((3/2)*log(z)^2 - polylog(2, 1-z) + polylog(2, 1-z^2)), where z = (-1 + (44 - 3*sqrt(177))^(1/3) + (44 + 3*sqrt(177))^(1/3))/6 = 0.82948354095849703967... is the real root of the equation z^3*(1 - z)/(1 - z^2)^2 = 1 and c = 0.57862299312... - Vaclav Kotesovec, Jun 29 2019, updated Oct 09 2024

G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 2*x^9 + 4*x^10 + 4*x^11 + 4*x^12 + 4*x^13 + 5*x^14 + 6*x^15 + 6*x^16 + 6*x^17 + ...

such that, by definition,

A(x) = 1 + x*(1 + x) + x^2*(1 + x^2)*(1 + x^3) + x^3*(1 + x^3)*(1 + x^4)*(1 + x^5) + x^4*(1 + x^4)*(1 + x^5)*(1 + x^6)*(1 + x^7) + x^5*(1 + x^5)*(1 + x^6)*(1 + x^7)*(1 + x^8)*(1 + x^9) + ... + x^n*Product_{k=0..n-1} (1 + x^(n+k)) + ...

Also

A(x) = 1/(1 - x) + x^2/((1 - x^2)*(1 - x^3)) + x^7/((1 - x^3)*(1 - x^4)*(1 - x^5)) + x^15/((1 - x^4)*(1 - x^5)*(1 - x^6)*(1 - x^7)) + x^26/((1 - x^5)*(1 - x^6)*(1 - x^7)*(1 - x^8)*(1 - x^9)) + ... + x^(n*(3*n+1)/2)/(Product_{k=0..n} 1 - x^(n+k+1)) + ...

With[{m = 80}, CoefficientList[Series[Sum[x^n*Product[1+x^(n+j), {j, 0, n-1}], {n, 0, m}], {x, 0, m}], x]] (* G. C. Greubel, Jan 12 2019 *)

nmax = 100; pk = x + x^2; s = 1 + pk; Do[pk = Normal[Series[pk * x*(1 + x^(2*k - 2))*(1 + x^(2*k - 1))/(1 + x^(k - 1)), {x, 0, nmax}]]; s = s + pk, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Jun 18 2019 *)

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

for(n=0, 80, print1(a(n), ", "))

(Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); [1] cat Coefficients(R!( (&+[x^n*(&*[1+x^(n+j): j in [0..n-1]]) : n in [1..m]]) )); // G. C. Greubel, Jan 12 2019

(SageMath)

R = PowerSeriesRing(ZZ, 'x')

m = 80

x = R.gen().O(m)

s = sum(x^n*prod(1+x^(n+j) for j in (0..n-1)) for n in (0..m))

s.coefficients() # G. C. Greubel, Jan 12 2019

Cf. A053263, A087135, A241061.

Sequence in context: A263020 A068796 A154804 * A053276 A064065 A284486

Adjacent sequences: A207639 A207640 A207641 * A207643 A207644 A207645

Paul D. Hanna, Feb 19 2012

approved