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A207642 Expansion of g.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + x^(n+k)).

%I A207642 #43 Oct 09 2024 04:19:44
%S A207642 1,1,2,1,2,2,2,3,3,2,4,4,4,4,5,6,6,6,6,8,9,8,10,10,10,12,14,14,14,15,
%T A207642 16,19,20,20,22,24,24,26,28,30,34,34,35,38,40,42,46,50,50,54,58,60,63,
%U A207642 66,70,76,80,84,88,92,96,102,108,112,120,126,131,140,146,151
%N A207642 Expansion of g.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + x^(n+k)).
%C A207642 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
%H A207642 Vaclav Kotesovec, <a href="/A207642/b207642.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Paul D. Hanna)
%F A207642 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
%F A207642 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
%e A207642 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 + ...
%e A207642 such that, by definition,
%e A207642 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)) + ...
%e A207642 Also
%e A207642 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)) + ...
%t A207642 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 *)
%t A207642 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 *)
%o A207642 (PARI) {a(n)=polcoeff(sum(m=0,n,x^m*prod(k=0,m-1,1+x^(m+k) +x*O(x^n))),n)}
%o A207642 for(n=0,80,print1(a(n),", "))
%o A207642 (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
%o A207642 (Sage)
%o A207642 R = PowerSeriesRing(ZZ, 'x')
%o A207642 m = 80
%o A207642 x = R.gen().O(m)
%o A207642 s = sum(x^n*prod(1+x^(n+j) for j in (0..n-1)) for n in (0..m))
%o A207642 s.coefficients() # _G. C. Greubel_, Jan 12 2019
%Y A207642 Cf. A053263, A087135, A241061.
%K A207642 nonn
%O A207642 0,3
%A A207642 _Paul D. Hanna_, Feb 19 2012