A092313 -id:A092313 - 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

A092268 Total number of smallest parts in all partitions of n into odd parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 12, 15, 20, 29, 36, 46, 61, 74, 95, 122, 145, 180, 224, 268, 328, 399, 474, 567, 682, 807, 955, 1136, 1330, 1564, 1842, 2140, 2499, 2914, 3375, 3917, 4533, 5220, 6014, 6929, 7942, 9102, 10430, 11898, 13582, 15489, 17600, 19999, 22706, 25719

Offset: 1

Vladeta Jovovic, Feb 16 2004

			Partitions of 6 into odd parts are: [1,1,1,1,1,1], [1,1,1,3], [3,3], [1,5]; thus a(6)=6+3+2+1=12.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2500

Cf. A092314, A092322, A092269, A092309, A092321, A092313, A092310, A092311.

Cf. A067588.

nmax = 50; Rest[CoefficientList[Series[Sum[(x^(2*n - 1)/(1 - x^(2*n - 1))) / Product[(1 - x^(2*k - 1)), {k, n, nmax}], {n, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 06 2019 *)

G.f.: Sum((x^(2*n-1)/(1-x^(2*n-1)))/Product((1-x^(2*k-1)), k=n..infinity), n=1..infinity).

a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)). - Vaclav Kotesovec, Jul 07 2019

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

A092311 Total number of largest parts in all partitions of n into odd parts.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 12, 14, 19, 23, 26, 33, 38, 44, 56, 63, 71, 88, 99, 114, 138, 155, 176, 208, 237, 269, 314, 357, 402, 468, 529, 594, 686, 772, 873, 999, 1119, 1260, 1431, 1608, 1804, 2039, 2284, 2554, 2884, 3219, 3590, 4032, 4493, 5011, 5603, 6231, 6928

Offset: 1

Vladeta Jovovic, Feb 16 2004

			Partitions of 6 into odd parts are: [1,1,1,1,1,1], [1,1,1,3], [3,3], [1,5]; thus a(6)=6+1+2+1=10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2500

Cf. A092314, A092322, A092269, A092309, A092321, A092313, A092310, A092268.

nmax = 50; Rest[CoefficientList[Series[Sum[(x^(2*n - 1)/(1 - x^(2*n - 1))) / Product[(1 - x^(2*k - 1)), {k, 1, n}], {n, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 06 2019 *)
lpp[k_]:=Module[{c=Max[k]},Count[k,c]]; Table[Total[lpp/@Select[IntegerPartitions[ n],AllTrue[ #,OddQ]&]],{n, 60}] (* Harvey P. Dale, Apr 24 2023 *)

G.f.: Sum((x^(2*n-1)/(1-x^(2*n-1)))/Product((1-x^(2*k-1)), k=1..n), n=1..infinity).

a(n) ~ exp(Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 07 2019

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

A092309 Sum of smallest parts (counted with multiplicity) of all partitions of n.

Original entry on oeis.org

1, 4, 7, 15, 19, 39, 46, 80, 106, 160, 201, 318, 390, 554, 729, 998, 1262, 1727, 2168, 2894, 3670, 4749, 5963, 7737, 9635, 12232, 15257, 19206, 23727, 29723, 36509, 45296, 55512, 68292, 83298, 102079, 123805, 150697, 182254, 220790, 265766

Offset: 1

Vladeta Jovovic, Feb 16 2004

			Partitions of 4 are: [1,1,1,1], [1,1,2], [2,2], [1,3], [4]; thus a(4)=4*1+2*1+2*2+1*1+1*4=15.

Alois P. Heinz, Table of n, a(n) for n = 1..9000

Cf. A092314, A092322, A092269, A092321, A092313, A092310, A092311, A092268.

Cf. A046746.

b:= proc(n, i) option remember; `if`(irem(n, i)=0, n, 0)
       +`if`(i>1, add(b(n-i*j, i-1), j=0..(n-1)/i), 0)
    end:
a:= n-> b(n$2):
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 04 2016

ss[n_]:=Module[{m=Min[n]},Select[n,#==m&]]; Table[Total[Flatten[ss/@ IntegerPartitions[n]]],{n,50}] (* Harvey P. Dale, Dec 16 2013 *)
b[n_, i_] := b[n, i] = If[Mod[n, i] == 0, n, 0] + If[i > 1, Sum[b[n - i*j, i - 1], {j, 0, (n - 1)/i}], 0]; a[n_] := b[n, n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

G.f.: Sum(n*x^n/(1-x^n)*Product(1/(1-x^k), k = n .. infinity), n = 1 .. infinity).

a(n) ~ sqrt(2) * exp(Pi*sqrt(2*n/3)) / (4*Pi*sqrt(n)). - Vaclav Kotesovec, Jul 06 2019

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

A092321 Sum of largest parts (counted with multiplicity) of all partitions of n.

Original entry on oeis.org

0, 1, 4, 8, 17, 26, 49, 69, 115, 164, 249, 343, 513, 686, 974, 1314, 1806, 2382, 3232, 4208, 5597, 7244, 9456, 12118, 15687, 19899, 25422, 32079, 40589, 50796, 63805, 79303, 98817, 122179, 151145, 185820, 228598, 279476, 341807, 416051, 506205, 613244, 742720

Offset: 0

Vladeta Jovovic, Feb 16 2004

			Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4]; thus a(4) = 4*1 + 1*2 + 2*2 + 1*3 + 1*4 = 17.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Margaret Archibald, A. Blecher, C. Brennan, A. Knopfmacher and T. Mansour, Partitions according to multiplicities and part sizes, Australasian Journal of Combinatorics, Volume 66(1) (2016), Pages 104-119.
Ljuben Mutafchiev, On the Largest Part Size and Its Multiplicity of a Random Integer Partition, arXiv:1712.03233 [math.PR], 2017.

Cf. A006128, A092314, A092322, A092269, A092309, A092313, A092310, A092311, A092268.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0$2], b(n, i-1, t) +add((l->`if`(t, l,
       l+[0, l[1]*i*j]))(b(n-i*j, i-1, true)), j=1..n/i)))
    end:
a:= n-> b(n$2, false)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2014

f[n_] := Block[{c = 2n, k = 2, p = IntegerPartitions[n]}, m = Max @@@ p; l = Length[p]; While[k < l, c = c + m[[k]]*Count[p[[k]], m[[k]]]; k++ ]; If[n == 1, 1, c]]; Table[ f[n], {n, 41}] (* Robert G. Wilson v, Feb 18 2004, updated by Jean-François Alcover, Jan 29 2014 *)
nmax = 50; CoefficientList[Series[Sum[n*x^n/(1-x^n) * Product[1/(1 - x^k), {k, 1, n}], {n, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 06 2019 *)
Join[{0},Table[Total[Flatten[First[Split[#]]&/@IntegerPartitions[n]]],{n,50}]] (* Harvey P. Dale, Oct 29 2019 *)

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

More terms from Robert G. Wilson v, Feb 18 2004

A092314 Sum of smallest parts of all partitions of n into odd parts.

Original entry on oeis.org

1, 1, 4, 2, 7, 6, 11, 8, 18, 16, 24, 23, 34, 36, 51, 48, 66, 74, 90, 98, 126, 137, 164, 182, 220, 247, 294, 324, 380, 434, 496, 556, 650, 728, 835, 938, 1068, 1204, 1372, 1531, 1736, 1956, 2198, 2462, 2784, 3104, 3482, 3890, 4358, 4864, 5441, 6048, 6748, 7516

Offset: 1

Vladeta Jovovic, Feb 15 2004

a(n) = Sum_{k>=1} k*A116856(n,k). - Emeric Deutsch, Feb 24 2006

			a(5)=7 because the partitions of 5 into odd parts are [5],[3,1,1] and [1,1,1,1,1] and the smallest parts add up to 5+1+1=7.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2500

Cf. A092322, A092269, A092309, A092321, A092313, A092310, A092311, A092268.

Cf. A116856, A092322.

g:=sum((2*n-1)*x^(2*n-1)/Product(1-x^(2*k-1),k=n..30),n=1..30): gser:=series(g,x=0,57): seq(coeff(gser,x^n),n=1..54); # Emeric Deutsch, Feb 24 2006

nmax = 50; Rest[CoefficientList[Series[Sum[(2*n - 1)*x^(2*n - 1) / Product[(1 - x^(2*k - 1)), {k, n, nmax}], {n, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 06 2019 *)

G.f.: Sum((2*n-1)*x^(2*n-1)/Product(1-x^(2*k-1), k = n .. infinity), n = 1 .. infinity).

a(n) ~ exp(Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 07 2019

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

A092322 Sum of largest parts of all partitions of n into odd parts.

Original entry on oeis.org

1, 1, 4, 4, 9, 12, 19, 24, 36, 48, 64, 83, 108, 140, 179, 224, 280, 352, 432, 532, 652, 795, 960, 1160, 1392, 1669, 1992, 2368, 2804, 3320, 3908, 4592, 5388, 6300, 7349, 8560, 9940, 11524, 13340, 15401, 17752, 20436, 23472, 26920, 30840, 35256, 40252, 45900

Offset: 1

Vladeta Jovovic, Feb 15 2004

a(n) = Sum_{k>=1} k*A116799(n,k). - Emeric Deutsch, Feb 24 2006

			a(5)=9 because the partitions of 5 into odd parts are [5],[3,1,1] and [1,1,1,1,1] and the largest parts add up to 5+3+1=9.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2500

Cf. A092314, A092269, A092309, A092321, A092313, A092310, A092311, A092268.

Cf. A116799.

g:=sum((2*n-1)*x^(2*n-1)/Product(1-x^(2*k-1),k=1..n),n=1..30): gser:=series(g,x=0,50): seq(coeff(gser,x^n),n=1..48); # Emeric Deutsch, Feb 24 2006

nmax = 50; Rest[CoefficientList[Series[Sum[(2*n - 1)*x^(2*n - 1) / Product[(1 - x^(2*k - 1)), {k, 1, n}], {n, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 06 2019 *)

G.f.: Sum_{n>=1} (2*n-1)*x^(2*n-1)/Product_{k=1..n} (1-x^(2*k-1)).

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

A092310 Sum of largest parts (counted with multiplicity) of all partitions of n into odd parts.

Original entry on oeis.org

1, 2, 6, 7, 13, 20, 28, 34, 53, 71, 88, 117, 148, 188, 250, 301, 365, 472, 565, 688, 860, 1027, 1224, 1486, 1771, 2107, 2524, 2983, 3496, 4158, 4867, 5666, 6676, 7762, 9021, 10525, 12145, 14034, 16249, 18696, 21478, 24721, 28308, 32364, 37110, 42289

Offset: 1

Vladeta Jovovic, Feb 16 2004

			Partitions of 6 into odd parts are: [1,1,1,1,1,1], [1,1,1,3], [3,3], [1,5]; thus a(6)=6*1+1*3+2*3+1*5=20.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2500

Cf. A092314, A092322, A092269, A092309, A092321, A092313, A092311, A092268.

nmax = 50; Rest[CoefficientList[Series[Sum[(2*n - 1)*x^(2*n - 1)/(1 - x^(2*n - 1)) / Product[(1 - x^(2*k - 1)), {k, 1, n}], {n, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 06 2019 *)

G.f.: Sum((2*n-1)*x^(2*n-1)/(1-x^(2*n-1))/Product(1-x^(2*k-1), k = 1 .. n), n = 1 .. infinity).

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

A090794 Number of partitions of n such that the number of different parts is odd.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 4, 9, 13, 19, 27, 43, 54, 71, 102, 124, 161, 200, 257, 319, 400, 484, 618, 761, 956, 1164, 1450, 1806, 2226, 2741, 3367, 4137, 5020, 6163, 7485, 9042, 10903, 13172, 15721, 18956, 22542, 26925, 31935, 37962, 44861, 53183, 62651

Offset: 1

Vladeta Jovovic, Feb 12 2004

			n=6 has A000041(6)=11 partitions: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1 and 1+1+1+1+1+1 with partition sets: {6}, {1,5}, {2,4}, {1,4}, {3}, {1,2,3}, {1,3}, {2}, {1,2}, {1,2} and {1}, five of them have an odd number of elements, therefore a(6)=5.

Cf. A060177, A002134, A000005, A027193, A090794.
Cf. also A092314, A092322, A092269, A092309, A092321, A092313, A092310, A092311, A092268
Cf. A092306.

import Data.List (group)
a090794 = length . filter odd . map (length . group) . ps 1 where
   ps x 0 = [[]]
   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 19 2013

a(n) = b(n, 1, 0, 0) with b(n, i, j, f) = if iReinhard Zumkeller, Feb 19 2004
G.f.: F(x)*G(x)/2, where F(x) = 1-Product(1-2*x^i, i=1..infinity) and G(x) = 1/Product(1-x^i, i=1..infinity).
a(n) = (A000041(n)-A104575(n))/2.
G.f. A(x) equals the off-diagonal entries in the 2 X 2 matrix Product_{n >= 1} [1, x^n/(1 - x^n); x^n/(1 - x^n), 1] = [B(x), A(x); A(x), B(x)], where B(x) is the g.f. of A092306. - Peter Bala, Feb 10 2021

More terms from Reinhard Zumkeller, Feb 17 2004

Definition simplified and shortened by Jonathan Sondow, Oct 13 2013

cp's OEIS Frontend

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

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A092268 Total number of smallest parts in all partitions of n into odd parts.

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A092311 Total number of largest parts in all partitions of n into odd parts.

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A092309 Sum of smallest parts (counted with multiplicity) of all partitions of n.

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A092321 Sum of largest parts (counted with multiplicity) of all partitions of n.

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A092314 Sum of smallest parts of all partitions of n into odd parts.

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A092322 Sum of largest parts of all partitions of n into odd parts.

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