A039899 -id:A039899 - cp's OEIS frontend

A003114 Number of partitions of n into parts 5k+1 or 5k+4.

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41, 46, 54, 61, 70, 79, 91, 102, 117, 131, 149, 167, 189, 211, 239, 266, 299, 333, 374, 415, 465, 515, 575, 637, 709, 783, 871, 961, 1065, 1174, 1299, 1429, 1579, 1735, 1913, 2100, 2311, 2533, 2785

Offset: 0

N. J. A. Sloane and Herman P. Robinson

Expansion of Rogers-Ramanujan function G(x) in powers of x.
Same as number of partitions into distinct parts where the difference between successive parts is >= 2.
As a formal power series, the limit of polynomials S(n,x): S(n,x)=sum(T(i,x),0<=i<=n); T(i,x)=S(i-2,x).x^i; T(0,x)=1,T(1,x)=x; S(n,1)=A000045(n+1), the Fibonacci sequence. - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001
The Rogers-Ramanujan identity is 1 + Sum_{n >= 1} t^(n^2)/((1-t)*(1-t^2)*...*(1-t^n)) = Product_{n >= 1} 1/((1-t^(5*n-1))*(1-t^(5*n-4))).
Coefficients in expansion of permanent of infinite tridiagonal matrix:
  1 1 0 0 0 0 0 0 ...
  x 1 1 0 0 0 0 0 ...
  0 x^2 1 1 0 0 0 ...
  0 0 x^3 1 1 0 0 ...
  0 0 0 x^4 1 1 0 ...
  ................... - Vladeta Jovovic, Jul 17 2004
Also number of partitions of n such that the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Jul 17 2004
Also number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Feb 27 2006
Also number of partitions of n such that if k is the largest part, then k occurs at least k times. Example: a(9)=5 because we have [3, 3, 3], [2, 2, 2, 2, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1]. - Emeric Deutsch, Apr 16 2006
a(n) = number of NW partitions of n, for n >= 1; see A237981.
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[1](x). - N. J. A. Sloane, Nov 22 2015
Convolution of A109700 and A109697. - Vaclav Kotesovec, Jan 21 2017

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
G.f. = 1/q + q^59 + q^119 + q^179 + 2*q^239 + 2*q^299 + 3*q^359 + 3*q^419 + ...
From _Joerg Arndt_, Dec 27 2012: (Start)
The a(16)=17 partitions of 16 where all parts are 1 or 4 (mod 5) are
  [ 1]  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
  [ 2]  [ 4 1 1 1 1 1 1 1 1 1 1 1 1 ]
  [ 3]  [ 4 4 1 1 1 1 1 1 1 1 ]
  [ 4]  [ 4 4 4 1 1 1 1 ]
  [ 5]  [ 4 4 4 4 ]
  [ 6]  [ 6 1 1 1 1 1 1 1 1 1 1 ]
  [ 7]  [ 6 4 1 1 1 1 1 1 ]
  [ 8]  [ 6 4 4 1 1 ]
  [ 9]  [ 6 6 1 1 1 1 ]
  [10]  [ 6 6 4 ]
  [11]  [ 9 1 1 1 1 1 1 1 ]
  [12]  [ 9 4 1 1 1 ]
  [13]  [ 9 6 1 ]
  [14]  [ 11 1 1 1 1 1 ]
  [15]  [ 11 4 1 ]
  [16]  [ 14 1 1 ]
  [17]  [ 16 ]
The a(16)=17 partitions of 16 where successive parts differ by at least 2 are
  [ 1]  [ 7 5 3 1 ]
  [ 2]  [ 8 5 3 ]
  [ 3]  [ 8 6 2 ]
  [ 4]  [ 9 5 2 ]
  [ 5]  [ 9 6 1 ]
  [ 6]  [ 9 7 ]
  [ 7]  [ 10 4 2 ]
  [ 8]  [ 10 5 1 ]
  [ 9]  [ 10 6 ]
  [10]  [ 11 4 1 ]
  [11]  [ 11 5 ]
  [12]  [ 12 3 1 ]
  [13]  [ 12 4 ]
  [14]  [ 13 3 ]
  [15]  [ 14 2 ]
  [16]  [ 15 1 ]
  [17]  [ 16 ]
(End)

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109, 238.
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(e), p. 591.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 669.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 107.
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 90-92.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 290-291.
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
George E. Andrews; R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 14.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy)
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
P. Jacob and P. Mathieu, Parafermionic derivation of Andrews-type multiple-sums, arXiv:hep-th/0505097, 2005.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, The Ramanujan Journal 29.1-3 (2012): 199-211.
I. Martinjak, D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), Research Paper 13, 122 pp.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities.
Mingjia Yang, Doron Zeilberger, Systematic Counting of Restricted Partitions, arXiv:1910.08989 [math.CO], 2019.

Cf. A003106, A003116, A127836, A003113, A006141, A039899, A039900.
Cf. A188216 (least part k occurs at least k times).
Cf. A047209, A203776, A237981.
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Row sums of A268187.

a003114 = p a047209_list where
   p _      0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jan 05 2011

a003114 = p 1 where
   p _ 0 = 1
   p k m = if k > m then 0 else p (k + 2) (m - k) + p (k + 1) m
-- Reinhard Zumkeller, Feb 19 2013

g:=sum(x^(k^2)/product(1-x^j,j=1..k),k=0..10): gser:=series(g,x=0,65): seq(coeff(gser,x,n),n=0..60); # Emeric Deutsch, Feb 27 2006

CoefficientList[ Series[Sum[x^k^2/Product[1 - x^j, {j, 1, k}], {k, 0, 10}], {x, 0, 65}], x][[1 ;; 61]] (* Jean-François Alcover, Apr 08 2011, after Emeric Deutsch *)
Table[Count[IntegerPartitions[n], p_ /; Min[p] >= Length[p]], {n, 0, 24}] (* Clark Kimberling, Feb 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 0, 0, -1, 0}[[Mod[k, 5, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 17 2015 *)
nmax = 60; kmax = nmax/5;
s = Flatten[{Range[0, kmax]*5 + 1}~Join~{Range[0, kmax]*5 + 4}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

{a(n) = my(t); if( n<0, 0, t = 1 + x * O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= x^(2*k - 1) / (1 - x^k) * (1 + x * O(x^(n - k^2))), 1), n))}; /* Michael Somos, Oct 15 2008 */

G.f.: Sum_{k>=0} x^(k^2)/(Product_{i=1..k} 1-x^i).
The g.f. above is the special case D=2 of sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ), the g.f. for partitions into distinct part where the difference between successive parts is >= D. - Joerg Arndt, Mar 31 2014
G.f.: 1 + sum(i=1, oo, x^(5i+1)/prod(j=1 or 4 mod 5 and j<=5i+1, 1-x^j) + x^(5i+4)/prod(j=1 or 4 mod 5 and j<=5i+4, 1-x^j)). - Jon Perry, Jul 06 2004
G.f.: (Product_{k>0} 1 + x^(2*k)) * (Sum_{k>=0} x^(k^2) / (Product_{i=1..k} 1 - x^(4*i))). - Michael Somos, Oct 19 2006
Euler transform of period 5 sequence [ 1, 0, 0, 1, 0, ...]. - Michael Somos, Oct 15 2008
Expansion of f(-x^5) / f(-x^1, -x^4) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, May 17 2015
Expansion of f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 13 2015
a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2 * 3^(1/4) * 5^(1/2) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(60*sqrt(15))) / sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 23 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017

A006141 Number of integer partitions of n whose smallest part is equal to the number of parts.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 25, 29, 33, 38, 42, 49, 54, 62, 69, 78, 87, 99, 109, 123, 137, 154, 170, 191, 211, 236, 261, 290, 320, 357, 392, 435, 479, 530, 582, 644, 706, 779, 854, 940, 1029, 1133, 1237, 1358, 1485

Offset: 1

N. J. A. Sloane

nonn

Or, number of partitions of n in which number of largest parts is equal to the largest part.
a(n) is the number of partitions of n-1 without parts that differ by less than 2 and which have no parts less than three. [MacMahon]
There are two conflicting choices for the offset in this sequence. For the definition given here the offset is 1, and that is what we shall adopt. On the other hand, if one arrives at this sequence via the Rogers-Ramanujan identities (see the next comment), the natural offset is 0.
Related to Rogers-Ramanujan identities: Let G[1](q) and G[2](q) be the generating functions for the two Rogers-Ramanujan identities of A003114 and A003106, starting with the constant term 1. The g.f. for the present sequence is G[3](q) = (G[1](q) - G[2](q))/q = 1+q^3+q^4+q^5+q^6+q^7+2*q^8+2*q^9+3*q^10+.... - Joerg Arndt, Oct 08 2012; N. J. A. Sloane, Nov 18 2015
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[3](x). - N. J. A. Sloane, Nov 22 2015
From Wolfdieter Lang, Oct 31 2016: (Start)
From Hardy (H) p. 94, eq. (6.12.1) and Hardy-Wright (H-W), p. 293,  eq. (19.14.3) for H_2(a,x) - H_1(a,x) = a*H_1(a*x,x) one finds from the result for H_1(a,x) (in (H) on top on p. 95), after putting a=x, the o.g.f. of a(n) = A003114(n) - A003106(n), n >= 0, with a(0) = 0 as Sum_{m>=0} x^((m+1)^2) / Product_{j=1..m} (1 - x^j). The m=0 term is  1*x^1. See the formula given by Joerg Arndt, Jan 29 2011.
This formula has a combinatorial interpretation (found similar to the one given in (H) section 6.0, pp. 91-92 or (H-W) pp. 290-291): a(n) is the number of partitions of n with parts differing by at least 2 and part 1 present. See the example for a(15) below. (End)
The Heinz numbers of these integer partitions are given by A324522. - Gus Wiseman, Mar 09 2019

			G.f. = x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 3*x^12 + ...
a(15) = 5 because the partitions of 15 where the smallest part equals the number of parts are
3 + 6 + 6,
3 + 5 + 7,
3 + 4 + 8,
3 + 3 + 9, and
2 + 13.
- _Joerg Arndt_, Oct 08 2012
a(15) = 5 because the partitions of 15 with parts differing by at least 2 and part 1 present are: [14,1] obtained from the partition of 11 with one part, [11], added to the first part of the special partition [3,1] of 4 and  [11,3,1], [10,4,1], [9,5,1], [8,6,1] from adding all partition of 15 - 9 = 6 with one part, [6], and those with two parts, [5,1], [4,1], [3,3], to the special partition [5,3,1] of 9. - _Wolfdieter Lang_, Oct 31 2016
a(15) = 5 because the partitions of 14 with parts >= 3 and parts differing by at least 2 are [14], [11,3], [10,4], [9,5] and [8,6]. See the second [MacMahon] comment. This follows from the g.f. G[3](q) given in Andrews - Baxter, eq. (5.1) for i=3, (using summation index  m) and  m*(m+2) = 3 + 5 + ... + (2*m+1). - _Wolfdieter Lang_, Nov 02 2016
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(8) = 1 through a(15) = 5 integer partitions:
  (6,2)  (7,2)    (8,2)    (9,2)    (10,2)   (11,2)   (12,2)   (13,2)
         (3,3,3)  (4,3,3)  (4,4,3)  (5,4,3)  (5,5,3)  (6,5,3)  (6,6,3)
                           (5,3,3)  (6,3,3)  (6,4,3)  (7,4,3)  (7,5,3)
                                             (7,3,3)  (8,3,3)  (8,4,3)
                                                               (9,3,3)
(End)

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 92-95.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 292-294.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 45, Section 293.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
George E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, arXiv:1205.6570 [math.CO], 2012; The Ramanujan Journal 29.1-3 (2012): 199-211.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Cf. A064174, A090858, A324516, A324518, A324520, A324522.
A003106 counts partitions with minimum > length.
A003114 counts partitions with minimum >= length.
A026794 counts partitions by minimum.
A039899 counts partitions with minimum < length.
A039900 counts partitions with minimum <= length.
A239950 counts partitions with minimum equal to number of distinct parts.
Sequences related to balance:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 ranks balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A055396, A114638, A117409, A325134, A340601, A340611, A340654, A340655.

b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i,i)))))
    end:
a:= n-> add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 08 2012

b[n_, i_] := b[n, i] = If[n<0, 0, If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]]; a[n_] := Sum[b[n-j^2, j-1], {j, 0, Sqrt[n]}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Min[#]==Length[#]&]],{n,30}] (* Gus Wiseman, Mar 09 2019 *)

{a(n) = if( n<1, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(j=1, k-1, 1 - x^j, 1 + O(x ^ (n - k^2 + 1) ))), n))} /* Michael Somos, Jan 22 2008 */

G.f.: Sum_{m>=1} (x^(m^2)-x^(m*(m+1))) / Product_{i=1..m} (1-x^i) .
G.f.: Sum_{n>=1} x^(n^2)/Product_{k=1..n-1} (1-x^k). - Joerg Arndt, Jan 29 2011
a(n) = A003114(n) - A003106(n) = A039900(n) - A039899(n), (offset 1). - Vladeta Jovovic, Jul 17 2004
Plouffe in his 1992 dissertation conjectured that this has g.f. = (1+z+z^4+2*z^5-z^3-z^8+3*z^10-z^7+z^9)/(1+z-z^4-2*z^3-z^8+z^10), but Michael Somos pointed out on Jan 22 2008 that this is false.
Expansion of ( f(-x^2, -x^3) - f(-x, -x^4) ) / f(-x) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 22 2007
a(n) ~ sqrt(1/sqrt(5) - 2/5) * exp(2*Pi*sqrt(n/15)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Nov 01 2016

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000
Better description from Naohiro Nomoto, Feb 06 2002
Name shortened by Gus Wiseman, Apr 07 2021 (balanced partitions are A047993).

A039900 Number of partitions satisfying 0 < cn(0,5) + cn(1,5) + cn(4,5).

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 9, 13, 19, 27, 38, 52, 71, 95, 127, 167, 220, 285, 370, 474, 607, 770, 976, 1226, 1540, 1920, 2391, 2960, 3660, 4501, 5529, 6760, 8254, 10038, 12190, 14750, 17825, 21470, 25825, 30975, 37101, 44322, 52879, 62937, 74811, 88733, 105110, 124261

Offset: 0

Olivier Gérard

nonn

For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: o < 0 + 1 + 4 (OMZAAp).
Number of partitions of n such that (greatest part) >= (multiplicity of greatest part), for n >= 1.  For example, a(6) counts these 9 partitions:  6, 51, 42, 411, 33, 321, 3111, 22111, 21111.  See the Mathematica program at A240057 for the sequence as a count of these partitions, along with counts of related partitions.  - Clark Kimberling, Apr 02 2014
The Heinz numbers of these integer partitions are given by A324561. - Gus Wiseman, Mar 09 2019
From Gus Wiseman, Mar 09 2019: (Start)
Also the number of integer partitions of n whose minimum part is less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324560. For example, the a(1) = 1 through a(7) = 13 integer partitions are:
  (1)  (11)  (21)   (22)    (32)     (42)      (52)
             (111)  (31)    (41)     (51)      (61)
                    (211)   (221)    (222)     (322)
                    (1111)  (311)    (321)     (331)
                            (2111)   (411)     (421)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

			From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(7) = 13 integer partitions with at least one part equal to 0, 1, or 4 modulo 5:
  (1)  (11)  (21)   (4)     (5)      (6)       (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (321)     (331)
                            (2111)   (411)     (421)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A003106, A003114, A039899.
Cf. A003114, A006141, A047993, A064174, A117144.
Cf. A324518, A324520, A324522, A324560, A324561.

b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i<1, 0, b(n, i-1, t)+ `if`(i>n, 0, b(n-i, i,
      `if`(irem(i, 5) in {2, 3}, t, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 03 2014

Table[Count[IntegerPartitions[n], p_ /; Min[p] <= Length[p]], {n, 40}] (* Clark Kimberling, Feb 13 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, t, If[i<1, 0, b[n, i-1, t] + If[i > n, 0, b[n-i, i, If[MemberQ[{2, 3}, Mod[i, 5]], t, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^k*(1-x^k^2)/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: Sum_{k>=0} x^k * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000041(n) - A003106(n). - Vaclav Kotesovec, Oct 20 2024

A237976 Number of strict partitions of n such that (least part) < number of parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 14, 17, 21, 25, 31, 37, 45, 54, 64, 76, 90, 106, 124, 146, 170, 198, 230, 267, 308, 357, 410, 472, 542, 621, 709, 811, 923, 1051, 1194, 1355, 1534, 1738, 1962, 2215, 2497, 2812, 3161, 3553, 3986, 4469, 5005, 5600, 6258

Offset: 0

Clark Kimberling, Feb 18 2014

			a(8) = 3 counts these partitions:  71, 521, 431.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A237977, A096401, A237979, A025157, A039899.

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^(k*(k-1)))/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k*(k-1))) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000009(n) - A025157(n). - Vaclav Kotesovec, Jan 18 2022

Prepended a(0)=0, Seiichi Manyama, Jan 13 2022

A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).

Original entry on oeis.org

0, 2, 3, 4, 6, 10, 14, 21, 28, 40, 53, 74, 97, 131, 171, 225, 290, 377, 480, 616, 779, 987, 1238, 1556, 1935, 2411, 2981, 3685, 4527, 5562, 6793, 8295, 10081, 12241, 14805, 17890, 21538, 25906, 31062, 37201, 44429, 53004, 63070, 74964, 88898, 105297

Offset: 1

Clark Kimberling, Apr 02 2014

Let # denote "number of" and c(p) = conjugate of partitionp.  Then
A240057(n) = # p such that min(p) not = max(c(p));
A039899(n) = # p such that min(p) < max(c(p));
A039900(n) = # p such that min(p) <= max(c(p));
A006141(n) = # p such that min(p) = max(c(p));
A003114(n) = # p such that min(p) > max(c(p));
A003016(n) = # p such that min(p) >= max(c(p));
A064173(n) = # p such that max(p) < max(c(p));
A064174(n) = # p such that max(p) <= max(c(p));
A047993(n) = # p such that max(p) = max(c(p)).
See A240178 for related sequences. - Clark Kimberling, Apr 11 2014

			a(9) = 28 counts all the 30 partitions of 9 except 333 and 2211111.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003106, A003114, A006141, A039899, A039900.

b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))))
    end:
a:= n->combinat[numbpart](n)-add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 03 2014

z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Max[p]]], {n, 0, z}]  (* A003106 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Max[p]]], {n, 0, z}] (* A003114 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Max[p]]], {n, 0, z}] (* A006141 *)
tt = Table[Count[f[n], p_ /; Max[p] != Count[p, Max[p]]], {n, 0, z}] (* A240057 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Max[p]]], {n, 0, z}] (* A039899 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Max[p]]], {n, 0, z}] (* A039900 *)
(* second program: *)
b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]];
a[n_] := PartitionsP[n] - Sum[b[n - j^2, j - 1], {j, 0, Sqrt[n]}];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) + A006141(n) = A000041(n) for n > 0.

cp's OEIS Frontend

A003114 Number of partitions of n into parts 5k+1 or 5k+4.

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A006141 Number of integer partitions of n whose smallest part is equal to the number of parts.

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A039900 Number of partitions satisfying 0 < cn(0,5) + cn(1,5) + cn(4,5).

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A237976 Number of strict partitions of n such that (least part) < number of parts.

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A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).

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A248374 The integer partition a(n) (compare A194602) has only the non-one addends n+1 and 2.

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