A102462 -id:A102462 - cp's OEIS frontend

A027193 Number of partitions of n into an odd number of parts.

0, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 29, 37, 52, 66, 90, 113, 151, 190, 248, 310, 400, 497, 632, 782, 985, 1212, 1512, 1851, 2291, 2793, 3431, 4163, 5084, 6142, 7456, 8972, 10836, 12989, 15613, 18646, 22316, 26561, 31659, 37556, 44601, 52743, 62416, 73593, 86809, 102064, 120025, 140736

Offset: 0

Clark Kimberling

nonn

Number of partitions of n in which greatest part is odd.
Number of partitions of n+1 into an even number of parts, the least being 1. Example: a(5)=4 because we have [5,1], [3,1,1,1], [2,1,1] and [1,1,1,1,1,1].
Also number of partitions of n+1 such that the largest part is even and occurs only once. Example: a(5)=4 because we have [6], [4,2], [4,1,1] and [2,1,1,1,1]. - Emeric Deutsch, Apr 05 2006
Also the number of partitions of n such that the number of odd parts and the number of even parts have opposite parities.  Example: a(8)=10 is a count of these partitions: 8, 611, 521, 431, 422, 41111, 332, 32111, 22211, 2111111. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
In Chaves 2011 see page 38 equation (3.20). - Michael Somos, Dec 28 2014
Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 10*x^8 + 16*x^9 + 20*x^10 + ...
From _Gus Wiseman_, Feb 11 2021: (Start)
The a(1) = 1 through a(8) = 10 partitions into an odd number of parts are the following. The Heinz numbers of these partitions are given by A026424.
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)
            (111)  (211)  (221)    (222)    (322)      (332)
                          (311)    (321)    (331)      (422)
                          (11111)  (411)    (421)      (431)
                                   (21111)  (511)      (521)
                                            (22111)    (611)
                                            (31111)    (22211)
                                            (1111111)  (32111)
                                                       (41111)
                                                       (2111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part is odd are the following. The Heinz numbers of these partitions are given by A244991.
  (1)  (11)  (3)    (31)    (5)      (33)      (7)        (53)
             (111)  (1111)  (32)     (51)      (52)       (71)
                            (311)    (321)     (322)      (332)
                            (11111)  (3111)    (331)      (521)
                                     (111111)  (511)      (3221)
                                               (3211)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (32111)
                                                          (311111)
                                                          (11111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 39, Example 7.

T. D. Noe, Table of n, a(n) for n = 0..999
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
D. R. C. Chaves, Um estudo combinatório e comparativo de identidades teta parciais de Andrews e Ramanujan, 2011. In Portuguese.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_0(n).

Cf. A000041, A000700, A000701, A026837, A046682.
The Heinz numbers of these partitions are A026424 or A244991.
The even-length version is A027187.
The case of odd sum as well as length is A160786, ranked by A340931.
The case of odd maximum as well as length is A340385.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A067659 counts strict partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
A000009 counts partitions into odd parts, ranked by A066208.
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A101707 counts partitions of odd positive rank.
Cf. A078408, A174725, A236914, A300272, A340831.

g:=sum(x^(2*k)/product(1-x^j,j=1..2*k-1),k=1..40): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..45); # Emeric Deutsch, Apr 05 2006

nn=40;CoefficientList[Series[ Sum[x^(2j+1)Product[1/(1- x^i),{i,1,2j+1}],{j,0,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Dec 01 2012 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n], OddQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n], OddQ[ First@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n + 1], #[[-1]] == 1 && EvenQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n + 1], EvenQ[ First@#] && (Length[#] < 2 || #[[1]] != #[[2]]) &]]; (* Michael Somos, Dec 28 2014 *)

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, if( k%2, x^k / prod( j=1, k, 1 - x^j, 1 + x * O(x^(n-k)) ))), n))}; /* Michael Somos, Jul 24 2012 */

q='q+O('q^66); concat([0], Vec( (1/eta(q)-eta(q)/eta(q^2))/2 ) ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) - (-1)^n*A000700(n)) / 2.
For g.f. see under A027187.
G.f.: Sum(k>=1, x^(2*k-1)/Product(j=1..2*k-1, 1-x^j ) ). - Emeric Deutsch, Apr 05 2006
G.f.: - Sum(k>=1, (-x)^(k^2)) / Product(k>=1, 1-x^k ). - Joerg Arndt, Feb 02 2014
G.f.: Sum(k>=1, x^(k*(2*k-1)) / Product(j=1..2*k, 1-x^j)). - Michael Somos, Dec 28 2014
a(2*n) = A000701(2*n), a(2*n-1) = A046682(2*n-1); a(n) = A000041(n)-A027187(n). - Reinhard Zumkeller, Apr 22 2006

A027187 Number of partitions of n into an even number of parts.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 40, 49, 69, 86, 118, 146, 195, 242, 317, 392, 505, 623, 793, 973, 1224, 1498, 1867, 2274, 2811, 3411, 4186, 5059, 6168, 7427, 9005, 10801, 13026, 15572, 18692, 22267, 26613, 31602, 37619, 44533, 52815, 62338, 73680, 86716, 102162, 119918

Offset: 0

Clark Kimberling

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]
Number of partitions of n+1 into an odd number of parts, the least being 1.
Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

			G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...
From _Gus Wiseman_, Jan 05 2021: (Start)
The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by A028260.
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (2211)    (2221)    (71)
                              (3111)    (3211)    (2222)
                              (111111)  (4111)    (3221)
                                        (211111)  (3311)
                                                  (4211)
                                                  (5111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)
The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by A244990.
  (2)  (21)  (4)    (41)    (6)      (43)      (8)
             (22)   (221)   (42)     (61)      (44)
             (211)  (2111)  (222)    (421)     (62)
                            (411)    (2221)    (422)
                            (2211)   (4111)    (431)
                            (21111)  (22111)   (611)
                                     (211111)  (2222)
                                               (4211)
                                               (22211)
                                               (41111)
                                               (221111)
                                               (2111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
George E. Andrews and David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 13.
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000701, A046682, A026838, A102462.
The Heinz numbers of these partitions are A028260.
The odd version is A027193.
The strict case is A067661.
The case of even sum as well as length is A236913 (the even bisection).
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A000009 counts partitions into odd parts, ranked by A066208.
A026805 counts partitions whose least part is even.
A072233 counts partitions by sum and length.
A101708 counts partitions of even positive rank.
Cf. A000700, A026424, A058696, A096373, A244990, A300061.

f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* Clark Kimberling, Mar 13 2012 *)
a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* Michael Somos, May 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* Michael Somos, Aug 19 2006 */

my(q='q+O('q^66)); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) + (-1)^n * A000700(n))/2.
a(n) = p(n) - p(n-1) + p(n-4) - p(n-9) + ... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan, Mar 14 2004
From Bill Gosper, Jun 25 2005: (Start)
G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3*q^4 + 3*q^5 + 6*q^6 + ...
  = Sum_{n >= 0} q^(2*n)/(q; q)_{2*n}
  = ((Product_{k >= 1} 1/(1-q^k)) + (Product_{k >= 1} 1/(1+q^k)))/2.
Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 5*q^6 + ...
Then B(q) = Sum_{n >= 0} q^(2*n+1)/(q; q){2*n+1} = ((Product{k >= 1} 1/(1-q^k)) - (Product_{k >= 1} 1/(1+q^k)))/2.
Also we have the following identity involving 2 X 2 matrices:
Product_{k >= 1} [ 1/(1-q^(2*k)), q^k/(1-q^(2*k)) ; q^k/(1-q^(2*k)), 1/(1-q^(2*k)) ]
  = [ A(q), B(q) ; B(q), A(q) ]. (End)
a(2*n) = A046682(2*n), a(2*n+1) = A000701(2*n+1); a(n) = A000041(n)-A027193(n). - Reinhard Zumkeller, Apr 22 2006
Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - Michael Somos, Aug 19 2006
G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - Michael Somos, Aug 19 2006
a(n) = A338914(n) + A096373(n). - Gus Wiseman, Jan 06 2021

Offset changed to 0 by Michael Somos, Jul 24 2012

A007294 Number of partitions of n into nonzero triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 7, 10, 11, 11, 15, 17, 17, 22, 24, 25, 32, 35, 36, 44, 48, 50, 60, 66, 68, 81, 89, 92, 107, 117, 121, 141, 153, 159, 181, 197, 205, 233, 252, 262, 295, 320, 332, 372, 401, 417, 465, 501, 520, 575, 619, 645, 710, 763

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

Also number of decreasing integer sequences l(1) >= l(2) >= l(3) >= .. 0 such that sum('i*l(i)','i'=1..infinity)=n.
a(n) is also the number of partitions of n such that #{parts equal to i} >= #{parts equal to j} if i <= j.
Also the number of partitions of n (necessarily into distinct parts) where the part sizes are monotonically decreasing (including the last part, which is the difference between the last part and a "part" of size 0). These partitions are the conjugates of the partitions with number of parts of size i increasing. - Franklin T. Adams-Watters, Apr 08 2008
Also partitions with condition as in A179255, and additionally, if more than one part, first difference >= first part: for example, a(10)=7 as there are 7 such partitions of 10:  1+2+3+4 = 1+2+7 = 1+3+6 = 1+9 = 2+8 = 3+7 = 10. - Joerg Arndt, Mar 22 2011
Number of members of A181818 with a bigomega value of n (cf. A001222). - Matthew Vandermast, May 19 2012

			6 = 3+3 = 3+1+1+1 = 1+1+1+1+1+1 so a(6) = 4.
a(7)=4: Four sequences as above are (7,0,..), (5,1,0,..), (3,2,0,..),(2,1,1,0,..). They correspond to the partitions 1^7, 2 1^5, 2^2 1^3, 3 2 1^2 of seven or in the main description to the partitions 1^7, 3 1^4, 3^2 1, 6 1.
From _Gus Wiseman_, May 03 2019: (Start)
The a(1) = 1 through a(9) = 6 partitions using nonzero triangular numbers are the following. The Heinz numbers of these partitions are given by A325363.
  1   11   3     31     311     6        61        611        63
           111   1111   11111   33       331       3311       333
                                3111     31111     311111     6111
                                111111   1111111   11111111   33111
                                                              3111111
                                                              111111111
The a(1) = 1 through a(10) = 7 partitions with weakly decreasing multiplicities are the following. Equivalent to Matthew Vandermast's comment, the Heinz numbers of these partitions are given by A025487 (products of primorial numbers).
  1  11  21   211   2111   321     3211     32111     32211      4321
         111  1111  11111  2211    22111    221111    222111     322111
                           21111   211111   2111111   321111     2221111
                           111111  1111111  11111111  2211111    3211111
                                                      21111111   22111111
                                                      111111111  211111111
                                                                 1111111111
The a(1) = 1 through a(11) = 7 partitions with weakly increasing differences (where the last part is taken to be zero) are the following. The Heinz numbers of these partitions are given by A325362 (A = 10, B = 11).
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)    (A)     (B)
            (21)  (31)  (41)  (42)   (52)   (62)   (63)   (73)    (83)
                              (51)   (61)   (71)   (72)   (82)    (92)
                              (321)  (421)  (521)  (81)   (91)    (A1)
                                                   (531)  (631)   (731)
                                                   (621)  (721)   (821)
                                                          (4321)  (5321)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006.
G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
N. A. Brigham, A General Asymptotic Formula for Partition Functions, Proc. Amer. Math. Soc., vol. 1 (1950), p. 191.
Jorge A. Campos-Gonzalez-Angulo, Raphael F. Ribeiro, and Joel Yuen-Zhou, Generalization of the Tavis-Cummings model for multi-level anharmonic systems, arXiv:2101.09475 [physics.optics], 2021.
Zhicheng Gao, Andrew MacFie and Daniel Panario, Counting words by number of occurrences of some patterns, The Electronic Journal of Combinatorics, 18 (2011), #P143.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, Journal of Integer Sequences, Vol. 7, 2004.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000217, A051533, A000294.
Cf. A102462.
Row sums of array A176723 and triangle A176724. - Wolfdieter Lang, Jul 19 2010
Cf. A179255 (condition only on differences), A179269 (parts strictly increasing instead of nondecreasing). - Joerg Arndt, Mar 22 2011
Cf. A024940, A280366.
Row sums of A319797.
Cf. A007862, A025487, A240026, A320509, A325324, A325354, A325356, A325362, A325363.

a007294 = p $ tail a000217_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, Jun 28 2013

b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=0 then 0
    else b(n, i-1) +b(n-i*(i+1)/2, i)
      fi
    end:
a:= n-> b(n, floor(sqrt(2*n))):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 22 2011
isNondecrP :=proc(L) slp := DIFF(DIFF(L)) ; min(op(%)) >= 0 ; end proc:
A007294 := proc(n) local a, p; a := 0 ; if n = 0 then return 1 ; end if; for p in combinat[partition](n) do if nops(p) = nops(convert(p, set)) then if isNondecrP(p) then if nops(p) =1 then a := a+1 ; elif op(2, p) >= 2*op(1, p) then a := a+1; end if; end if; end if; end do; a ; end proc:
seq(A007294(n), n=0..30) ; # R. J. Mathar, Jan 07 2011

CoefficientList[ Series[ 1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 70}], x]
(* also *)
t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)
b[n_, i_] := b[n, i] = Which[n < 0, 0, n == 0, 1, i == 0, 0, True, b[n, i-1]+b[n-i*(i+1)/2, i]]; a[n_] := b[n, Floor[Sqrt[2*n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Differences[Append[#,0]]]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)
nmax = 58; t = Table[PolygonalNumber[n], {n, nmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[t, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

N=66; Vec(1/prod(k=1,N,1-x^(k*(k+1)\2))+O(x^N)) \\ Joerg Arndt, Apr 14 2013

from functools import lru_cache
from sympy import divisors
from sympy.ntheory.primetest import is_square
@lru_cache(maxsize=None)
def A007294(n):
    @lru_cache(maxsize=None)
    def a(n): return is_square((n<<3)+1)
    @lru_cache(maxsize=None)
    def c(n): return sum(d for d in divisors(n,generator=True) if a(d))
    return (c(n)+sum(c(k)*A007294(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

def A007294(n):
    has_nondecreasing_diffs = lambda x: min(differences(x, 2)) >= 0
    special = lambda x: (x[1]-x[0]) >= x[0]
    allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_nondecreasing_diffs(x))
    return len([1 for x in Partitions(n, max_slope=-1) if allowed(x[::-1])]) # D. S. McNeil, Jan 06 2011

G.f.: 1/Product_{k>=2} (1-z^binomial(k, 2)).
For n>0: a(n) = b(n, 1) where b(n, k) = if n>k*(k+1)/2 then b(n-k*(k+1)/2, k) + b(n, k+1) else (if n=k*(k+1)/2 then 1 else 0). - Reinhard Zumkeller, Aug 26 2003
For n>0, a(n) is Euler Transform of [1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,...], i.e A010054, n>0. - Benedict W. J. Irwin, Jul 29 2016
a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2) / (2^(7/2) * sqrt(3) * Pi * n^(3/2)) [Brigham 1950 (exponential part), Almkvist 2006]. - Vaclav Kotesovec, Dec 31 2016
G.f.: Sum_{i>=0} x^(i*(i+1)/2) / Product_{j=1..i} (1 - x^(j*(j+1)/2)). - Ilya Gutkovskiy, May 07 2017

Additional comments from Roland Bacher, Jun 17 2001

A361216 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle.

Original entry on oeis.org

1, 1, 4, 2, 11, 56, 3, 29, 370, 5752, 4, 94, 2666, 82310, 2519124, 6, 263, 19126, 1232770, 88117873, 6126859968, 12, 968, 134902, 19119198, 2835424200

Offset: 1

Pontus von Brömssen, Mar 05 2023

Tilings that are rotations or reflections of each other are considered distinct.

Pieces can have any combination of integer side lengths, but for the optimal sets computed so far (up to (n,k) = (7,5)), all pieces have one side of length 1.

			Triangle begins:
  n\k|  1    2       3         4            5          6  7  8
  ---+--------------------------------------------------------
  1  |  1
  2  |  1    4
  3  |  2   11      56
  4  |  3   29     370      5752
  5  |  4   94    2666     82310      2519124
  6  |  6  263   19126   1232770     88117873 6126859968
  7  | 12  968  134902  19119198   2835424200          ?  ?
  8  | 20 3416 1026667 307914196 109979838540          ?  ?  ?
A 3 X 3 square can be tiled by three 1 X 2 pieces and three 1 X 1 pieces in the following ways:
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+   +   +---+---+   +---+   +---+
  |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +   +---+---+   +   +---+---+   +
  |       |   |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+   +---+---+---+
  |       |   |   |   |   |   |   |   |       |
  +---+---+---+   +---+---+   +   +---+---+---+
  |   |   |   |   |       |   |   |       |   |
  +---+---+   +   +---+---+---+   +---+---+---+
  |       |   |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+
  |       |   |   |       |   |
  +---+---+---+   +---+---+---+
  |       |   |   |   |       |
  +---+---+---+   +---+---+---+
  |       |   |   |       |   |
  +---+---+---+   +---+---+---+
The first six of these have no symmetries, so they account for 8 tilings each. The last two has a mirror symmetry, so they account for 4 tilings each. In total there are 6*8+2*4 = 56 tilings. This is the maximum for a 3 X 3 square, so T(3,3) = 56.
The following table shows the sets of pieces that give the maximum number of tilings up to (n,k) = (7,5). The solutions are unique except for (n,k) = (2,1) and (n,k) = (6,1).
      \     Number of pieces of size
  (n,k)\  1 X 1 | 1 X 2 | 1 X 3 | 1 X 4
  ------+-------+-------+-------+------
  (1,1) |   1   |   0   |   0   |   0
  (2,1) |   2   |   0   |   0   |   0
  (2,1) |   0   |   1   |   0   |   0
  (2,2) |   2   |   1   |   0   |   0
  (3,1) |   1   |   1   |   0   |   0
  (3,2) |   2   |   2   |   0   |   0
  (3,3) |   3   |   3   |   0   |   0
  (4,1) |   2   |   1   |   0   |   0
  (4,2) |   4   |   2   |   0   |   0
  (4,3) |   3   |   3   |   1   |   0
  (4,4) |   5   |   4   |   1   |   0
  (5,1) |   3   |   1   |   0   |   0
  (5,2) |   4   |   3   |   0   |   0
  (5,3) |   4   |   4   |   1   |   0
  (5,4) |   7   |   5   |   1   |   0
  (5,5) |   7   |   6   |   2   |   0
  (6,1) |   2   |   2   |   0   |   0
  (6,1) |   1   |   1   |   1   |   0
  (6,2) |   4   |   4   |   0   |   0
  (6,3) |   7   |   4   |   1   |   0
  (6,4) |   8   |   5   |   2   |   0
  (6,5) |  10   |   7   |   2   |   0
  (6,6) |  11   |   8   |   3   |   0
  (7,1) |   2   |   1   |   1   |   0
  (7,2) |   5   |   3   |   1   |   0
  (7,3) |   8   |   5   |   1   |   0
  (7,4) |  10   |   6   |   2   |   0
  (7,5) |  11   |   7   |   2   |   1

Main diagonal: A361217.
Columns: A102462 (k = 1), A361218 (k = 2), A361219 (k = 3), A361220 (k = 4).
Cf. A360629, A361221.

T(n,1) = A102462(n).

A072811 T(n,k) = multiplicity of the k-th partition of n in Mathematica order, defined to be the count of its permutations (compositions).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 3, 1, 6, 4, 1, 6, 5, 1, 1, 2, 2, 3, 2, 6, 4, 3, 3, 12, 5, 4, 10, 6, 1, 1, 2, 2, 3, 2, 6, 4, 1, 6, 3, 12, 5, 3, 6, 12, 20, 6, 1, 10, 15, 7, 1, 1, 2, 2, 3, 2, 6, 4, 2, 6, 3, 12, 5, 3, 6, 12, 12, 20, 6, 1, 12, 10, 4, 30, 30, 7, 5, 20, 21, 8, 1

Offset: 0

Wouter Meeussen, Aug 09 2002

The sum of row n equals A011782(n). The first and last columns equal 1. The number of integers per row equals the partition number p(n). Row n is a vector of weights or multiplicities relating counts of ordered versus unordered objects classified according to the partitions of n.
a(n) is the multinomial coefficient of the signature of the n-th partition. - Franklin T. Adams-Watters, Apr 08 2008
Let f(x)=1/(1-sum(j>=1, c[j]*x^j))=sum(n>=0, w(n)*x^n), then the coefficients of wn=Pn(c[1],...,c[n]), listed in reverse lexicographic order, give row n of T(n,k). - Groux Roland, Mar 08 2011

			The partitions of 4 are {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1}, so the fourth row equals 1,2,1,3,1 since these are the counts of the permutations of these lists.
Triangle begins:
1;
1;
1, 1;
1, 2, 1;
1, 2, 1, 3, 1;
1, 2, 2, 3, 3, 4, 1;
1, 2, 2, 3, 1, 6, 4, 1, 6, 5, 1;

Alois P. Heinz, Rows n = 0..26, flattened

Cf. A080577, A080575, A115621, A102462.

mult[li:{__Integer}] := Apply[Multinomial, Length/@Split[ Sort[li] ] ]; Table[mult/@Partitions[n], {n, 12}]

\\ here mulp(v) computes the multiplicity of the given partition.
mulp(v) = {my(p=(#v)!, k=1); for(i=2, #v, k=if(v[i]==v[i-1], k+1, p/=k!; 1)); p/k!}
Row(n)={apply(mulp, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 9, print(Row(n))) } \\ Peter Dolland, Nov 11 2019

cp's OEIS Frontend

A027193 Number of partitions of n into an odd number of parts.

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A027187 Number of partitions of n into an even number of parts.

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A007294 Number of partitions of n into nonzero triangular numbers.

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A361216 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle.

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A072811 T(n,k) = multiplicity of the k-th partition of n in Mathematica order, defined to be the count of its permutations (compositions).

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A102463 a(n) is the number of distinct values of (Sum_{i=1..r} x_i)!/(Product_{i=1..r} x_i!), where (x_1, ..., x_r) is an r-tuple of nonnegative integers with Sum_{i=1..r} i*x_i = n.

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A102465 a(n) = number of distinct values of Product_{i=1..r} x_i!*i!^x_i, where (x_1, ..., x_r) is an r-tuple of nonnegative integers with Sum_{i=1..r} i*x_i = n.

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A102465 a(n) = number of distinct values of Product_{i=1..r} x_i!i!^x_i, where (x_1, ..., x_r) is an r-tuple of nonnegative integers with Sum_{i=1..r} ix_i = n.