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%I #30 Jan 09 2022 09:59:03
%S 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 264,313,370,436,512,600,702,818,952,1106,1282,1484,1715,1978,2278,
%U 2620,3008,3448,3948,4512,5150,5872,6684,7600,8632,9791,11094,12558,14198,16036,18096,20398
%N Number of basis partitions (or basic partitions) of n.
%C 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.
%H Vaclav Kotesovec, <a href="/A066447/b066447.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 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 G.f.: sum(n>=0, x^(n^2) * prod(k=1..n, (1+x^k)/(1-x^k) ) ) [Given in Nolan et al. reference]. [_Joerg Arndt_, Apr 07 2011]
%F 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 a(n) ~ c * d^sqrt(n) / n^(3/4), where d = 7.1578741786143524880205016499891016... and c = 0.193340468476900308848561788251945... - _Vaclav Kotesovec_, Mar 19 2020
%p 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 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 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 (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 Row sums of A066448.
%Y Cf. A001130, A001935, A003114, A306734, A333374.
%K nonn,easy
%O 0,3
%A Herbert S. Wilf, Dec 29 2001
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