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A066447 Number of basis partitions (or basic partitions) of n. 2
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

Table of n, a(n) for n=0..55.

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]

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);

PROG

(PARI)

N=66; 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 07 2011 */

CROSSREFS

Row sums of A066448. Cf. A001130.

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 November 13 02:59 EST 2019. Contains 329085 sequences. (Running on oeis4.)