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A092269
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Spt function: total number of smallest parts in all partitions of n.
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53
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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
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OFFSET
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1,2
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COMMENTS
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Row sums of triangle A220504. - Omar E. Pol, Jan 19 2013
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LINKS
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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
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
K. Ono, Congruences for the Andrews spt-function
O. E. Pol, Illustration of initial terms
Wikipedia, Spt function
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FORMULA
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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*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
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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-François Alcover, Jan 17 2013 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A092314, A092322, A092309, A092321, A092313, A092310, A092311, A092268, A006128, A195053.
For higher-order spt functions see A221140-A221144.
Sequence in context: A078411 A137630 A220489 * A182722 A089483 A190484
Adjacent sequences: A092266 A092267 A092268 * A092270 A092271 A092272
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KEYWORD
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nonn
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AUTHOR
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Vladeta Jovovic, Feb 16 2004
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004
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STATUS
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approved
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