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A066447
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Number of basis partitions (or basic partitions) of n.
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6
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1, 1, 2, 2, 3, 4, 6, 8, 10, 13, 16, 20, 26, 32, 40, 50, 61, 74, 90, 108, 130, 156, 186, 222, 264, 313, 370, 436, 512, 600, 702, 818, 952, 1106, 1282, 1484, 1715, 1978, 2278, 2620, 3008, 3448, 3948, 4512, 5150, 5872, 6684, 7600, 8632, 9791, 11094, 12558, 14198, 16036, 18096, 20398
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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George E. Andrews, Partition Identities for Two-Color Partitions, 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.
J. M. Nolan, C. D. Savage and H. S. Wilf, Basis partitions, Discrete Math. 179 (1998), 277-283.
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FORMULA
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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]
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
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = 7.1578741786143524880205016499891016... and c = 0.193340468476900308848561788251945... - Vaclav Kotesovec, Mar 19 2020
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MAPLE
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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);
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MATHEMATICA
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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 *)
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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Herbert S. Wilf, Dec 29 2001
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STATUS
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approved
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