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 A066447 Number of basis partitions (or basic partitions) of n. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 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. FORMULA 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 MAPLE 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); MATHEMATICA 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 *) PROG (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 */ CROSSREFS Row sums of A066448. Cf. A001130, A001935, A003114, A306734, A333374. Sequence in context: A077114 A118246 A116902 * A035542 A225484 A130081 Adjacent sequences: A066444 A066445 A066446 * A066448 A066449 A066450 KEYWORD nonn,easy AUTHOR Herbert S. Wilf, Dec 29 2001 STATUS approved

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Last modified February 24 10:37 EST 2024. Contains 370294 sequences. (Running on oeis4.)