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A092269 Spt function: total number of smallest parts in all partitions of n. 56
1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589, 801, 1048, 1407, 1820, 2399, 3087, 3998, 5092, 6545, 8263, 10486, 13165, 16562, 20630, 25773, 31897, 39546, 48692, 59960, 73423, 89937, 109553, 133439, 161840, 196168, 236843, 285816, 343667, 412950, 494702, 592063, 706671 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums of triangle A220504. - Omar E. Pol, Jan 19 2013

LINKS

Joerg Arndt, Table of n, a(n) for n = 1..550

F. G. Garvan, Table of a(n) for n=1..10000 (Coefficients of Andrews spt-function)

G. E. Andrews, The number of smallest parts in the partitions of n

Scott Ahlgren, Nickolas Andersen, Euler-like recurrences for smallest parts functions, arXiv:1402.5366

George E. Andrews, Song Heng Chan and Byungchan Kim, The Odd Moments of Ranks and Cranks, 2012. - From N. J. A. Sloane, Sep 04 2012

G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function

G. E. Andrews, F. G. Garvan, and J. Liang, Self-conjugate vector partitions and the parity of the spt-function

A. Folsom and K. Ono, The spt-function of Andrews

F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank

F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13

F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences

F. G. Garvan, Higher Order Spt-functions, Adv. Math. 228 (2011), no. 1, 241-265; . - From N. J. A. Sloane, Jan 02 2013

F. G. Garvan, The smallest parts partition function, 2012

F. G. Garvan, Dyson's rank function and Andrews's SPT-function, slides 11, 12.

K. Ono, Congruences for the Andrews spt-function

Omar E. Pol, Illustration of initial terms

Wikipedia, Spt function

FORMULA

G.f.: sum(n>=1, x^n/(1-x^n) * prod(k>=n, 1/(1-x^k) ).

a(n) = A000070(n-1) + A195820(n). - Omar E. Pol, Oct 19 2011

a(n) = n*p(n) - N_2(n)/2 = n*A000041(n) - A220908(n)/2 = A066186(n) - A220907(n) = (A220909(n) - A220908(n))/2 = A211982(n)/2. (from Andrews's paper and Garvan's paper). - Omar E. Pol, Jan 03 2013

a(n) = A000041(n) + A000070(n-2) + A220479(n), n>=2. - Omar E. Pol, Feb 16 2013

EXAMPLE

Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4]. 1 appears 4 times in the first, 1 twice in the second, 2 twice in the third, etc.; thus a(4)=4+2+2+1+1=10.

MAPLE

b:= proc(n, i) option remember; `if`(n=0 or i=1, n,

      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))

    end:

a:= n-> b(n, n):

seq (a(n), n=1..60);  # Alois P. Heinz, Jan 16 2013

MATHEMATICA

terms = 47; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest (* Jean-Fran├žois Alcover, Jan 17 2013 *)

PROG

(PARI)

N = 66;  x = 'x + O('x^N);

gf = sum(n=1, N, x^n/(1-x^n) * prod(k=n, N, 1/(1-x^k) )  );

v = Vec(gf)

/* Joerg Arndt, Jan 12 2013 */

CROSSREFS

Cf. A092314, A092322, A092309, A092321, A092313, A092310, A092311, A092268, A006128, A195053.

For higher-order spt functions see A221140-A221144.

Sequence in context: A137630 A220489 A229915 * A182722 A089483 A190484

Adjacent sequences:  A092266 A092267 A092268 * A092270 A092271 A092272

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Feb 16 2004

EXTENSIONS

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

STATUS

approved

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Last modified August 20 02:49 EDT 2014. Contains 245794 sequences.