cp's OEIS Frontend

A066447 Number of basis partitions (or basic partitions) of n.

%I A066447 #40 Jan 09 2025 19:07:57
%S A066447 1,1,2,2,3,4,6,8,10,13,16,20,26,32,40,50,61,74,90,108,130,156,186,222,
%T A066447 264,313,370,436,512,600,702,818,952,1106,1282,1484,1715,1978,2278,
%U A066447 2620,3008,3448,3948,4512,5150,5872,6684,7600,8632,9791,11094,12558,14198,16036,18096,20398
%N A066447 Number of basis partitions (or basic partitions) of n.
%C A066447 The k-th successive rank of a partition pi = (pi_1, pi_2, ..., pi_s) of the integer n is r_k = pi_k - pi'_k, where pi' denotes the conjugate partition. A partition pi is basic if the number of dots in its Ferrers diagram is the least among all the Ferrers diagrams of partitions with the same rank vector.
%C A066447 The g.f. Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) = Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 + x^k - 2*x^k)  ==  Sum_{n >= 0} x^(n^2)  (mod 2). It follows that a(n) is odd iff n is a square (Nolan et al., equation 6, p. 282). - _Peter Bala_, Jan 08 2025
%H A066447 Vaclav Kotesovec, <a href="/A066447/b066447.txt">Table of n, a(n) for n = 0..10000</a>
%H A066447 George E. Andrews, <a href="https://hal.archives-ouvertes.fr/hal-03498190/">Partition Identities for Two-Color Partitions</a>, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.
%H A066447 J. M. Nolan, C. D. Savage and H. S. Wilf, <a href="https://doi.org/10.1016/S0012-365X(97)00101-5">Basis partitions</a>, Discrete Math. 179 (1998), 277-283.
%F A066447 G.f.: Sum_{n >= 0} x^(n^2) * Product_{k = 1..n} (1 + x^k)/(1 - x^k) [Given in Nolan et al. reference]. - _Joerg Arndt_, Apr 07 2011
%F A066447 Limit_{n->infinity} a(n) / A333374(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... - _Vaclav Kotesovec_, Mar 17 2020
%F A066447 a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.193340468476900308848561788251945... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 + cosh(arccosh(53*sqrt(11/2)/64)/3) / (3*sqrt(22))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - _Vaclav Kotesovec_, Mar 19 2020, updated Oct 10 2024
%p A066447 b := proc(n,d); option remember; if n=0 and d=0 then RETURN(1) elif n<=0 or d<=0 then RETURN(0) else RETURN(b(n-d,d)+b(n-2*d+1,d-1)+b(n-3*d+1,d-1)) fi: end: A066447 := n->add(b(n,d),d=0..n);
%t A066447 nmax = 60; CoefficientList[Series[Sum[x^(n^2)*Product[(1 + x^k)/(1 - x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 17 2020 *)
%t A066447 nmax = 60; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k - 1), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* _Vaclav Kotesovec_, Mar 17 2020 *)
%o A066447 (PARI) N=66; x='x+O('x^N); s=sum(n=0,N,x^(n^2)*prod(k=1,n,(1+x^k)/(1-x^k))); Vec(s) /* _Joerg Arndt_, Apr 07 2011 */
%Y A066447 Row sums of A066448.
%Y A066447 Cf. A001130, A001935, A003114, A306734, A333374.
%Y A066447 Cf. A192918, A376841.
%K A066447 nonn,easy
%O A066447 0,3
%A A066447 Herbert S. Wilf, Dec 29 2001