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A092269 Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.

%I A092269 #152 Feb 01 2024 09:43:57
%S A092269 1,3,5,10,14,26,35,57,80,119,161,238,315,440,589,801,1048,1407,1820,
%T A092269 2399,3087,3998,5092,6545,8263,10486,13165,16562,20630,25773,31897,
%U A092269 39546,48692,59960,73423,89937,109553,133439,161840,196168,236843,285816,343667,412950,494702,592063,706671
%N A092269 Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.
%C A092269 Row sums of triangle A220504. - _Omar E. Pol_, Jan 19 2013
%H A092269 Alois P. Heinz, <a href="/A092269/b092269.txt">Table of n, a(n) for n = 1..10000</a> (first 550 terms from Joerg Arndt)
%H A092269 F. G. Garvan, <a href="https://qseries.org/fgarvan/data/sptcofs.txt">Table of a(n) for n = 1..10000</a> (Coefficients of Andrews spt-function)
%H A092269 G. E. Andrews, <a href="https://doi.org/10.1515/CRELLE.2008.083">The number of smallest parts in the partitions of n</a>, Journal für die reine und angewandte Mathematik, Volume 2008 Issue 624, 133-142.
%H A092269 Scott Ahlgren, Nickolas Andersen, <a href="https://arxiv.org/abs/1402.5366">Euler-like recurrences for smallest parts functions</a>, arXiv:1402.5366
%H A092269 George E. Andrews, Song Heng Chan and Byungchan Kim, <a href="https://doi.org/10.1016/j.jcta.2012.07.001">The Odd Moments of Ranks and Cranks</a>, Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91. - From _N. J. A. Sloane_, Sep 04 2012
%H A092269 G. E. Andrews, F. G. Garvan, and J. Liang, <a href="https://qseries.org/fgarvan/papers/spt-crank.pdf">Combinatorial interpretation of congruences for the spt-function</a>
%H A092269 G. E. Andrews, F. G. Garvan, and J. Liang, <a href="https://georgeandrews1.github.io/pdf/289.pdf">Self-conjugate vector partitions and the parity of the spt-function</a>
%H A092269 William Y.C. Chen, <a href="https://arxiv.org/abs/1707.04369">The spt-Function of Andrews</a>, arXiv:1707.04369 [math.CO], Jul 14 2017.
%H A092269 A. Folsom and K. Ono, <a href="https://doi.org/10.1073/pnas.0809431105">The spt-function of Andrews</a>, PNAS, December 23 2008, 105 (51) 20152-20156.
%H A092269 F. G. Garvan, <a href="https://qseries.org/fgarvan/papers/spt.pdf">Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank</a>
%H A092269 F. G. Garvan, <a href="https://arxiv.org/abs/1011.1955">Congruences for Andrews' spt-function modulo powers of 5, 7 and 13</a>, arXiv:1011.1955 [math.NT], Nov 9 2010.
%H A092269 F. G. Garvan, <a href="https://arxiv.org/abs/1011.1957">Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences</a>, arXiv:1011.1957 [math.NT], Nov 9 2010.
%H A092269 F. G. Garvan, <a href="https://doi.org/10.1016/j.aim.2011.05.013">Higher Order Spt-functions</a>, Adv. Math. 228 (2011), no. 1, 241-265. - From _N. J. A. Sloane_, Jan 02 2013
%H A092269 F. G. Garvan, <a href="https://carmamaths.org/meetings/alfcon/pdfs/Frank_Garvan-alfcon.pdf">The smallest parts partition function</a>, 2012.
%H A092269 F. G. Garvan, <a href="http://www.combinatorics.net/conf/A75/Slides/02_03_Garvan.pdf">Dyson's rank function and Andrews's SPT-function</a>, slides 11, 12.
%H A092269 M. H. Mertens, K. Ono, L. Rolen, <a href="https://arxiv.org/abs/1906.07410">Mock modular Eisenstein series with Nebentypus</a>, arXiv:1906.07410 [math.NT], 2019.
%H A092269 K. Ono, <a href="https://doi.org/10.1073/pnas.1015339107">Congruences for the Andrews spt-function</a>, PNAS, December 21 2010, 108 (2) 473-476.
%H A092269 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa407.jpg">Illustration of initial terms</a>
%H A092269 Wikipedia, <a href="https://en.wikipedia.org/wiki/Spt_function">Spt function</a>
%F A092269 G.f.: Sum_{n>=1} x^n/(1-x^n) * Product_{k>=n} 1/(1-x^k).
%F A092269 a(n) = A000070(n-1) + A195820(n). - _Omar E. Pol_, Oct 19 2011
%F A092269 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
%F A092269 a(n) = A000041(n) + A000070(n-2) + A220479(n), n >= 2. - _Omar E. Pol_, Feb 16 2013
%F A092269 Asymptotics (Bringmann-Mahlburg, 2009): a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*sqrt(8*n)) ~ sqrt(6*n)*A000041(n)/Pi. - _Vaclav Kotesovec_, Jul 30 2017
%e A092269 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.
%p A092269 b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
%p A092269       `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
%p A092269     end:
%p A092269 a:= n-> b(n, n):
%p A092269 seq(a(n), n=1..60);  # _Alois P. Heinz_, Jan 16 2013
%t A092269 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 *)
%t A092269 b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r==0, q, 0] + Sum[b[n-i*j, i-1], {j, 0, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Nov 23 2015, after _Alois P. Heinz_ *)
%o A092269 (PARI)
%o A092269 N = 66;  x = 'x + O('x^N);
%o A092269 gf = sum(n=1,N, x^n/(1-x^n) * prod(k=n,N, 1/(1-x^k) )  );
%o A092269 v = Vec(gf)
%o A092269 /* _Joerg Arndt_, Jan 12 2013 */
%Y A092269 Cf. A092314, A092322, A092309, A092321, A092313, A092310, A092311, A092268, A006128, A195053.
%Y A092269 For higher-order spt functions see A221140-A221144.
%K A092269 nonn
%O A092269 1,2
%A A092269 _Vladeta Jovovic_, Feb 16 2004
%E A092269 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004