A220504 -id:A220504 - cp's OEIS frontend

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

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

Offset: 1

Vladeta Jovovic, Feb 16 2004

nonn

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

			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.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 550 terms from Joerg Arndt)
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, Journal für die reine und angewandte Mathematik, Volume 2008 Issue 624, 133-142.
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, Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91. - 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
William Y.C. Chen, The spt-Function of Andrews, arXiv:1707.04369 [math.CO], Jul 14 2017.
A. Folsom and K. Ono, The spt-function of Andrews, PNAS, December 23 2008, 105 (51) 20152-20156.
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, arXiv:1011.1955 [math.NT], Nov 9 2010.
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], Nov 9 2010.
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.
M. H. Mertens, K. Ono, L. Rolen, Mock modular Eisenstein series with Nebentypus, arXiv:1906.07410 [math.NT], 2019.
K. Ono, Congruences for the Andrews spt-function, PNAS, December 21 2010, 108 (2) 473-476.
Omar E. Pol, Illustration of initial terms
Wikipedia, Spt function

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

For higher-order spt functions see A221140-A221144.

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

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 *)
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 *)

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 */

G.f.: Sum_{n>=1} x^n/(1-x^n) * Product_{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
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

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

A174455 Number of partitions where the number of 1's and 2's are equal.

Original entry on oeis.org

1, 0, 0, 2, 1, 1, 4, 3, 4, 8, 8, 10, 17, 18, 23, 34, 39, 48, 67, 78, 97, 127, 151, 185, 237, 281, 343, 428, 511, 616, 759, 902, 1084, 1315, 1562, 1863, 2242, 2649, 3147, 3752, 4424, 5222, 6190, 7266, 8545, 10062, 11776, 13782, 16157, 18832, 21964, 25622, 29777, 34589, 40200, 46556, 53912

Offset: 0

Joerg Arndt, Nov 28 2010

nonn

From Omar E. Pol, Jan 19 2013: (Start)
Column 3 of triangle A220504.
With offset 3, a(n) is also the number of appearances of 3 as the smallest part in all partitions of n.
Also consider the sequence formed by [0, 0] together with this sequence, with offset 1, then it appears that A027336(n) = Sum_{j=1..3} a(n+j), n >= 0.
(End)

			a(8)=9, there are 8 such partitions of 9, they are
  #1:    9 =  3* 1 + 3* 2 + 0    + 0    + 0    + 0    + 0    + 0    + 0
  #2:    9 =  2* 1 + 2* 2 + 1* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #3:    9 =  1* 1 + 1* 2 + 2* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #4:    9 =  0    + 0    + 3* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #5:    9 =  0    + 0    + 0    + 1* 4 + 1* 5 + 0    + 0    + 0    + 0
  #6:    9 =  1* 1 + 1* 2 + 0    + 0    + 0    + 1* 6 + 0    + 0    + 0
  #7:    9 =  0    + 0    + 1* 3 + 0    + 0    + 1* 6 + 0    + 0    + 0
  #8:    9 =  0    + 0    + 0    + 0    + 0    + 0    + 0    + 0    + 1* 9

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A008483, A182713, A240056.

b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
      else `if`(i=3 and irem(n, 3, 'r')=0, r, 0); for j from 0
      to n/i do %+b(n-i*j, i-1) od; % fi
    end:
a:= n-> b(n+3, n+3):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 20 2013

(* See A240056. - Clark Kimberling, Mar 31 2014 *)
m = 66; gf = 1/((1-x^3)*Product[1-x^n, {n, 3, m}]) + O[x]^m; CoefficientList[gf, x] (* Jean-François Alcover, Jul 02 2015, after Joerg Arndt *)

N=66;  x='x+O('x^N);
gf=1/( (1-x^3) * prod(n=3,N, 1-x^n) );
Vec(gf)
/* Joerg Arndt, Jul 07 2012 */

G.f.: 1/( (1-x^3) * Product_{n>=3} (1-x^n) ). - Joerg Arndt, Jul 07 2012
a(n) = A182713(n+2) - A182713(n) = A240056(n+1) - A240056(n) for n >= 0. - Clark Kimberling, Mar 31 2014
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (9 * 2^(3/2) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2022

A209314 Tetrahedron T(n,k,j) of appearances of k as the smallest parts in all partition of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 0, 0, 1, 1, 1, 2, 3, 1, 1, 0, 0, 0, 1, 1, 1, 2, 3, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 5, 7, 1, 1, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 5, 7, 11, 0, 1, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3

Offset: 1

Omar E. Pol, Jan 20 2013

The sum of row k of the slice n is A220504(n,k).
The sum of all elements of the n-th slice is spt(n) = A092269(n), the smallest parts partition function.
First row of slice n lists the first n partition numbers A000041.

			--------------------------------------------------
.                        Triangle
Tetrahedron              A220504       A092269(n)
(Slice n = 1..6)      (the row sums)     spt(n)
--------------------------------------------------
1;                           1             1
--------------------------------------------------
1, 1;                        2
1;                           1             3
--------------------------------------------------
1, 1, 2;                     4
0, 0;                        0
1;                           1             5
--------------------------------------------------
1, 1, 2, 3;                  7
1, 1, 0;                     2
0, 0;                        0
1;                           1            10
--------------------------------------------------
1, 1, 2, 3, 5;              12
0, 1, 0, 0;                  1
0, 0, 0;                     0
0, 0;                        0
1;                           1            14
--------------------------------------------------
1, 1, 2, 3, 5, 7;           19
1, 1, 2, 0, 0;               4
1, 1, 0, 0;                  2
0, 0, 0;                     0
0, 0;                        0
1;                           1            26
--------------------------------------------------

Cf. A000041, A092269, A220504.

A220484 Triangle read by rows: T(j,k) is the total number of appearances of the smallest parts in the j-th partition of n, with partitions as nonincreasing lists of parts in lexicographic order.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 1, 2, 1, 5, 3, 2, 1, 1, 1, 1, 6, 4, 3, 2, 2, 1, 1, 3, 1, 2, 1, 7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 1, 9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1

Offset: 1

Omar E. Pol, Jan 20 2013

The sum of row n equals spt(n) = A092269(n), the smallest part partition function.

			For n = 5:
------------------------------------------
.                         number of
Partitions of 5         smallest parts
------------------------------------------
1 + 1 + 1 + 1 + 1              5
2 + 1 + 1 + 1                  3
3 + 1 + 1                      2
2 + 2 + 1                      1
4 + 1                          1
3 + 2                          1
5                              1
------------------------------------------
So row 5 is [5, 3, 2, 1, 1, 1, 1]. The sum of row 5 is 5+3+2+1+1+1+1 = spt(5) = A092269(n) = 14.
.
Written as an irregular triangle begins:
1;
2,1;
3,1,1;
4,2,1,2,1;
5,3,2,1,1,1,1;
6,4,3,2,2,1,1,3,1,2,1;
7,5,4,3,3,2,2,1,1,1,1,2,1,1,1;
8,6,5,4,4,3,3,2,2,2,2,1,1,1,1,4,2,1,1,1,2,1;
9,7,6,5,5,4,4,3,3,3,3,2,2,2,2,1,1,1,1,1,1,1,3,2,1,1,3,1,1,1;

Column 1 is A000027. Row n has length A000041(n). Row sums give A092269.

Cf. A209514, A220504.

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

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A174455 Number of partitions where the number of 1's and 2's are equal.

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A209314 Tetrahedron T(n,k,j) of appearances of k as the smallest parts in all partition of n.

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A220484 Triangle read by rows: T(j,k) is the total number of appearances of the smallest parts in the j-th partition of n, with partitions as nonincreasing lists of parts in lexicographic order.

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