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A207642 Expansion of g.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + x^(n+k)). 4

%I #41 Sep 08 2022 08:46:01

%S 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 16,19,20,20,22,24,24,26,28,30,34,34,35,38,40,42,46,50,50,54,58,60,63,

%U 66,70,76,80,84,88,92,96,102,108,112,120,126,131,140,146,151

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

%C 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 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 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 a(n) ~ c * exp(r*sqrt(n)) / sqrt(n), where r = 0.926140105877... and c = 0.57862299312... - _Vaclav Kotesovec_, Jun 29 2019

%e 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 such that, by definition,

%e 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 Also

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

%o (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 (Sage)

%o R = PowerSeriesRing(ZZ, 'x')

%o m = 80

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

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

%o s.coefficients() # _G. C. Greubel_, Jan 12 2019

%Y Cf. A053263, A087135, A241061.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 19 2012

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Last modified August 1 01:16 EDT 2024. Contains 374809 sequences. (Running on oeis4.)