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A066447
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Number of basis partitions (or basic partitions) of n.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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.
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LINKS
| J. M. Nolan, C. D. Savage and H. S. Wilf, Basis partitions, Discrete Math. 179 (1998), 277-283.
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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 7 2011]
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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);
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PROG
| (Pari)
N=44; x='x+O('x^N); /* that many terms */
s=sum(n=0, N, x^(n^2)*prod(k=1, n, (1+x^k)/(1-x^k))); /* g.f. */
Vec(s) /* show terms */ /* Joerg Arndt, Apr 7 2011 */
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CROSSREFS
| Row sums of A066448. Cf. A001130.
Sequence in context: A077114 A118246 A116902 * A035542 A130081 A141847
Adjacent sequences: A066444 A066445 A066446 * A066448 A066449 A066450
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KEYWORD
| nonn,easy
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AUTHOR
| Herbert S. Wilf (wilf(AT)math.upenn.edu), Dec 29 2001
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