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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
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.
Sequence in context: A077114 A118246 A116902 * A035542 A225484 A130081
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.)