A072233 -id:A072233 - cp's OEIS frontend

A008284 Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number of partitions of n into k positive parts, 1 <= k <= n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1, 1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

N. J. A. Sloane

From Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010: (Start)
A000041(n+1) = 1 + Sum_{r=1..n} Sum_{k=1..min(r,n-r+1)} T(r,k).
T(n, n-k) is also the number of partitions of k in which the greatest part is at most n-k. (End)
From Richard R. Forberg, Dec 26 2014: (Start)
Elements of T(n, k) for n <= 2+3k, equal A000041(n-k) - A000070(n-2k-1), with the assumption A000070(n) = 0 for n < 0.
The diagonal T(2+2k, k), for k > 1 equals A007042, and the diagonal T(3+3k,k), for k >= 1, equals A104384. (End)
T(-n, k) is used as a definition for A380038, which can therefore be seen as an extension of this sequence for negative n. - Friedjof Tellkamp, Jan 18 2025

			The triangle T(n,k) begins:
   n\k 1  2  3  4  5  6  7  8  9 10 11 12 ...
   1:  1
   2:  1  1
   3:  1  1  1
   4:  1  2  1  1
   5:  1  2  2  1  1
   6:  1  3  3  2  1  1
   7:  1  3  4  3  2  1  1
   8:  1  4  5  5  3  2  1  1
   9:  1  4  7  6  5  3  2  1  1
  10:  1  5  8  9  7  5  3  2  1  1
  11:  1  5 10 11 10  7  5  3  2  1  1
  12:  1  6 12 15 13 11  7  5  3  2  1  1
... Reformatted and extended by _Wolfdieter Lang_, Dec 03 2012; additional extension by _Bob Selcoe_, Jun 09 2013
T(7,3) = 4 because we have [3,3,1], [3,2,2], [3,2,1,1] and [3,1,1,1,1], each having greatest part 3; or [5,1,1], [4,2,1], [3,3,1] and [3,2,2] each having 3 parts.
* Example from formula above: T(10,4) = 9 because T(6,4) + T(6,3) + T(6,2) + T(6,1) = 2 + 3 + 3 + 1 = 9.
* P(n) = P(n-1) + DT(n-1). P(n) = unordered partitions of n. (A000041) DT(n-1) = sum of diagonals beginning at T(n-1,1).
Example P(11) = 56, P(10) = 42, sum DT(10) = 1 + 4 + 5 + 3 + 1 = 14. - _Bob Selcoe_, Jun 09 2013
From _Omar E. Pol_, Nov 19 2019: (Start)
Illustration of initial terms: T(n,k) is also the number of vertical line segments in the k-th column of the n-th diagram, which represents the partitions of n:
.
    1    1 1    1 1 1    1 2 1 1    1 2 2 1 1    1 3 3 2 1 1    1 3 4 3 2 1 1
.
   _|   _| |   _| | |   _| | | |   _| | | | |   _| | | | | |   _| | | | | | |
        _ _|   _ _| |   _ _| | |   _ _| | | |   _ _| | | | |   _ _| | | | | |
               _ _ _|   _ _ _| |   _ _ _| | |   _ _ _| | | |   _ _ _| | | | |
                        _ _|   |   _ _|   | |   _ _|   | | |   _ _|   | | | |
                        _ _ _ _|   _ _ _ _| |   _ _ _ _| | |   _ _ _ _| | | |
                                   _ _ _|   |   _ _ _|   | |   _ _ _|   | | |
                                   _ _ _ _ _|   _ _ _ _ _| |   _ _ _ _ _| | |
                                                _ _|   |   |   _ _|   |   | |
                                                _ _ _ _|   |   _ _ _ _|   | |
                                                _ _ _|     |   _ _ _|     | |
                                                _ _ _ _ _ _|   _ _ _ _ _ _| |
                                                               _ _ _|   |   |
                                                               _ _ _ _ _|   |
                                                               _ _ _ _|     |
                                                               _ _ _ _ _ _ _|
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.
D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 3: Generating All Combinations and Partitions, Addison-Wesley Professional, 2005, pp. 38, 45, 50 [From Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010]
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 400.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 294.

Franklin T. Adams-Watters, First 200 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831. [scanned copy]
Henry Bottomley, Illustration of initial terms
D. J. Broadhurst and D. Kreimer, Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees, arXiv:hep-th/0001202, 2000.
FindStat - Combinatorial Statistic Finder, The length of the partition
Martin Griffiths, Generating Functions for Extended Stirling Numbers of the First Kind, Journal of Integer Sequences, 17 (2014), #14.6.4.
Roser Homs and Anna-Lena Winz, Deformations of local Artin rings via Hilbert-Burch matrices, arXiv:2309.06871 [math.AC], 2023. See p. 16.
Wolfdieter Lang, First 10 rows and more.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Tilman Piesk, Illustration of initial terms
Stephan Scholz and Lothar Berger, Fast computation of function composition derivatives for flatness-based control of diffusion problems, J. Math. Industry (2024) Vol. 14, Art. No. 15.
Teerapat Srichan, Watcharapon Pimsert, and Vichian Laohakosol, New Recursion Formulas for the Partition Function, J. Int. Seq., Vol. 24 (2021), Article 21.6.6.
Eric Weisstein's World of Mathematics, Partition Function P
Index entries for triangles generated by the Multiset Transformation

A000041 is row sums and diagonal.
Columns k=1-10 are: A057427, A004526, A069905, A026810, A026811, A026812, A026813, A026814, A026815, A026816.
Partial sums of rows gives A026820.
Cf. A038497, A038498, A039805-A039809, A060016, A126988, A380038.
Read from right to left gives A058398.
Subtriangle of A072233 without row n=0 and column m=0.
Cf. A007042, A104384 which are diagonals with slope -2, -3.

a008284 n k = a008284_tabl !! (n-1) !! (k-1)
a008284_row n = a008284_tabl !! (n-1)
a008284_tabl = [1] : f [[1]] where
   f xss = ys : f (ys : xss) where
     ys = (map sum $ zipWith take [1..] xss) ++ [1]
-- Reinhard Zumkeller, Sep 05 2014

G:=-1+1/product(1-t*x^j,j=1..15): Gser:=simplify(series(G,x=0,17)): for n from 1 to 14 do P[n]:=coeff(Gser,x^n) od: for n from 1 to 14 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form; Emeric Deutsch, Feb 12 2006
with(combstruct):for n from 0 to 18 do seq(count(Partition(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Mar 30 2009
T := proc(n,k) option remember; if k < 0 or n < 0 then 0 elif k = 0 then if n = 0 then 1 else 0 fi else T(n - 1, k - 1) + T(n - k, k) fi end: seq(print(seq(T(n, k), k=1..n)),n=1..14); # Peter Luschny, Jul 24 2011

Column[Table[ IntegerPartitions[n, {k}] // Length, {n, 1, 20}, {k, 1, n}], Center] (* Frederik Beaujean (beaujean(AT)mpp.mpg.de), Apr 09 2010 *)
(*Recurrence closely related to natural numbers and number of divisors of n*)
Clear[t]; nn = 14; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, n - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0];Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]][[1 ;; 96]] (* Mats Granvik, Jan 01 2015 *)
Table[SeriesCoefficient[1/QPochhammer[a q, q], {q, 0, n}, {a, 0, k}], {n, 1, 15}, {k, 1, n}] // Column (* Vladimir Reshetnikov, Nov 18 2016 *)
T[n_, k_] := T[n, k] = If[n>0 && k>0, T[n-1, k-1] + T[n-k, k], Boole[n==0 && k==0]]
Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Robert A. Russell, May 12 2018 after Knuth 7.2.1.4 (39) *)

T(n,k)=#partitions(n-k,k)
for(n=1,9,for(k=1,n,print1(T(n,k)", "))) \\ Charles R Greathouse IV, Jan 04 2016

A8284=[]; A008284(n,k)={for(n=#A8284+1,n,A8284=concat(A8284,[vector(n,k,if(2*k1,A8284[n-k][k]+A8284[n-1][k-1],1),numbpart(n-k)))]));if(k,A8284[n][k],A8284[n])} \\ Without 2nd argument, return row n. - M. F. Hasler, Sep 26 2017

from functools import lru_cache
@lru_cache(maxsize=None)
def A008284_T(n,k):
    if k==n or k==1: return 1
    if k>n: return 0
    return A008284_T(n-1,k-1)+A008284_T(n-k,k) # Chai Wah Wu, Sep 21 2023

from sage.combinat.partition import number_of_partitions_length
[[number_of_partitions_length(n, k) for k in (1..n)] for n in (1..12)] # Peter Luschny, Aug 01 2015

T(n, k) = Sum_{i=1..k} T(n-k, i), for 1 <= k <= n-1; T(n, n) = 1 for n >= 1.
Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k > n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. for k-th column: x^k/(Product_{j=1..k} (1-x^j)). - Wolfdieter Lang, Nov 29 2000
G.f.: A(x, y) = Product_{n>=1} 1/(1-x^n)^(P_n(y)/n), where P_n(y) = Sum_{d|n} eulerphi(n/d)*y^d. - Paul D. Hanna, Jul 13 2004
If k >= n/2, T(n,k) = T(2(n-k),n-k) = A000041(n-k). - Franklin T. Adams-Watters, Jan 12 2006  [Relation included by Hans Loeblich, Apr 16 2019, relation extended by Evan Robinson, Jun 30 2021]
G.f.: G(t,x) = -1 + 1/Product_{j>=1} (1-t*x^j). - Emeric Deutsch, Feb 12 2006
A002865(n) = Sum_{k=2..floor((n+2)/2)} T(n-k+1,k-1). - Reinhard Zumkeller, Nov 04 2007
A000700(n) = Sum_{k=1..n} (-1)^(n-k) T(n,k). - Jeremy L. Martin, Jul 06 2013
G.f.: -1 + e^(F(x,z)), where F(x,z) = Sum_{n >= 1} (x*z)^n/(n*(1 - z^n)) is a g.f. for A126988. - Peter Bala, Jan 13 2015
Also, T(n, n-k) = k for k = 1, 2, 3; n >= 2k. T(n, 2) = floor(n/2). T(n, 3) = round(n^2/12). - M. F. Hasler, Sep 26 2017
T(n,k) = [n>0 & k>0] * (T(n-1,k-1) + T(n-k,k)) + [n==0 & k==0]. - Robert A. Russell, May 12 2018 from Knuth 7.2.1.4 (39)
T(n, k) = Sum_{i=0..n-1} T(n-ik-1, k-1) for k >= 1; T(-n, k) = 0 for n > 0; T(n, 0) = [n==0]. - Joshua Swanson (writing for Juexian Li), May 24 2020

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

Original entry on oeis.org

1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876, 203286806, 340435588, 568496753, 946695386

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g., let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.
Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g., for n = 4 we have:
  4   31   3   22   2   211   21   2    2   1111   111   11    11   1
           1        2         1    11   1          1     11    1    1
                                        1                      1    1
                                                                    1
Also total number of different species of singularity for conjugate functions with n letters (Sylvester).
According to [Belmans], this sequence gives "[t]he number of Segre symbols for the intersection of two quadrics in a fixed dimension". - Eric M. Schmidt, Sep 02 2017
From Gus Wiseman, Jul 30 2022: (Start)
Also the number of non-isomorphic multiset partitions of weight n with all constant blocks. The strict case is A089259. For example, non-isomorphic representatives of the a(1) = 1 through a(3) = 6 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{1},{1},{1}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
A000688 counts factorizations into prime powers.
A007716 counts non-isomorphic multiset partitions by weight.
A279784 counts twice-partitions of type PPR, factorizations A295935.
Constant partitions are ranked by prime-powers: A000961, A023894, A054685, A246655, A355743.
(End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + ...
a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).
The a(4)=14 multiset partitions whose total sum of parts is 4 are:
((4)),
((13)), ((1)(3)),
((22)), ((2)(2)),
((112)), ((1)(12)), ((2)(11)), ((1)(1)(2)),
((1111)), ((1)(111)), ((11)(11)), ((1)(1)(11)), ((1)(1)(1)(1)). - _Gus Wiseman_, Dec 19 2016

A. Cayley, Recherches sur les matrices dont les termes sont des fonctions linéaires d'une seule indéterminée, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order', Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 (first 500 terms from T. D. Noe)
Pieter Belmans, Segre symbols, 2016.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See pp. 10, 16.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).
L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, 2014.
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, Science China Physics, Mechanics & Astronomy, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9.
Jessie Pitsillides, Segre Characteristic Equivalence, arXiv:2506.12065 [math.GM], 2025.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
N. J. A. Sloane, Transforms.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
Index entries for sequences related to rooted trees

Related to A001383 via generating function.
The multiplicative version (factorizations) is A050336.
The ordered version (sequences of partitions) is A055887.
Row-sums of A061260.
Main diagonal of A055885.
We have A271619(n) <= a(n) <= A063834(n).
Column k=3 of A290353.
The strict case is A316980.
Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
Cf. A089300.
Cf. A001055, A072233, A112798, A275024.

Following Vladeta Jovovic:
a001970 n = a001970_list !! (n-1)
a001970_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y : ys) where
            y = sum (zipWith (*) ys a061259_list) `div` x
-- Reinhard Zumkeller, Oct 31 2015

with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
# second Maple program:
with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 19 2016

m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* Jean-François Alcover, Jul 19 2011, after g.f. *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k + x * O(x^n)), n))}; /* Michael Somos, Dec 20 2016 */

from sympy.core.cache import cacheit
from sympy import npartitions, divisors
@cacheit
def a(n): return 1 if n == 0 else sum([sum([d*npartitions(d) for d in divisors(j)])*a(n - j) for j in range(1, n + 1)]) / n
[a(n) for n in range(51)]  # Indranil Ghosh, Aug 19 2017, after Maple code
# (Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 1)
a = EulerTransform(EulerTransform(b))
print([a(n) for n in range(36)]) # Peter Luschny, Nov 17 2022

G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]
a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n > 1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic, Apr 21 2001
Logarithmic derivative yields A061259 (equivalent to above formula from Vladeta Jovovic). - Paul D. Hanna, Sep 05 2012
a(n) = Sum_{k=1..A000041(n)} A001055(A215366(n,k)) = number of factorizations of Heinz numbers of integer partitions of n. - Gus Wiseman, Dec 19 2016
a(n) = |{m>=1 : n = Sum_{k=1..A001222(m)} A056239(A112798(m,k)+1)}| = number of normalized twice-prime-factored multiset partitions (see A275024) whose total sum of parts is n. - Gus Wiseman, Dec 19 2016

Additional comments from Valery A. Liskovets

Sylvester references from Barry Cipra, Oct 07 2003

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

Original entry on oeis.org

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

A097805 Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read by rows for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 5, 1, 0, 1, 6, 15, 20, 15, 6, 1, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 1, 8, 28, 56, 70, 56, 28, 8, 1, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Paul Barry, Aug 25 2004

Previous name was: Riordan array (1, 1/(1-x)) read by rows.
Note this Riordan array would be denoted (1, x/(1-x)) by some authors.
Columns have g.f. (x/(1-x))^k. Reverse of A071919. Row sums are A011782. Antidiagonal sums are Fibonacci(n-1). Inverse as Riordan array is (1, 1/(1+x)). A097805=B*A059260*B^(-1), where B is the binomial matrix.
(0,1)-Pascal triangle. - Philippe Deléham, Nov 21 2006
(n+1) * each term of row n generates triangle A127952: (1; 0, 2; 0, 3, 3; 0, 4, 8, 4; ...). - Gary W. Adamson, Feb 09 2007
Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2008
From Paul Weisenhorn, Feb 09 2011: (Start)
Triangle read by rows: T(r,c) is the number of unordered partitions of n=r*(r+1)/2+c into (r+1) parts < (r+1) and at most pairs of equal parts and parts in neighboring pairs have difference 2.
Triangle read by rows: T(r,c) is the number of unordered partitions of the number n=r*(r+1)/2+(c-1) into r parts < (r+1) and at most pairs of equal parts and parts in neighboring pairs have difference 2. (End)
Triangle read by rows: T(r,c) is the number of ordered partitions (compositions) of r into c parts. - Juergen Will, Jan 04 2016
From Tom Copeland, Oct 25 2012: (Start)
Given a basis composed of a sequence of polynomials p_n(x) characterized by ladder (creation / annihilation, or raising / lowering) operators defined by R p_n(x) = p_(n+1)(x) and L p_n(x) = n p_(n-1)(x) with p_0(x)=1, giving the number operator # p_n(x) = RL p_n(x) = n p_n(x), the lower triangular padded Pascal matrix Pd (A097805) serves as a matrix representation of the operator exp(R^2*L) = exp(R#) =
  1) exp(x^2D) for the set x^n  and
  2) D^(-1) exp(t*x)D for the set x^n/n! (see A218234).
  (End)
From James East, Apr 11 2014: (Start)
Square array a(m,n) with m,n=0,1,2,... read by off-diagonals.
a(m,n) gives the number of order-preserving functions f:{1,...,m}->{1,...,n}. Order-preserving means that xa(n,n)=A088218(n) is the size of the semigroup O_n of all order-preserving transformations of {1,...,n}.
Read as a triangle, this sequence may be obtained by augmenting Pascal's triangle by appending the column 1,0,0,0,... on the left.
(End)
A formula based on the partitions of n with largest part k is given as a Sage program below. The 'conjugate' formula leads to A048004. - Peter Luschny, Jul 13 2015
From Wolfdieter Lang, Feb 17 2017: (Start)
The transposed of this lower triangular Riordan matrix of the associated type T provides the transition matrix between the monomial basis {x^n}, n >= 0, and the basis {y^n}, n >= 0, with y = x/(1-x): x^0 = 1 = y^0, x^n = Sum_{m >= n} Ttrans(n,m) y^m, for n >= 1, with Ttrans(n,m) = binomial(m-1,n-1).
Therefore, if a transformation with this Riordan matrix from a sequence {a} to the sequence {b} is given by b(n) = Sum_{m=0..n} T(n, m)*a(m), with T(n, m) = binomial(n-1, m-1), for n >= 1, then Sum_{n >= 0} a(n)*x^n = Sum_{n >= 0} b(n)*y^n, with y = x/(1-x) and vice versa. This is a modified binomial transformation; the usual one belongs to the Pascal Riordan matrix A007318. (End)
From Gus Wiseman, Jan 23 2022: (Start)
Also the number of compositions of n with alternating sum k, with k ranging from -n to n in steps of 2. For example, row n = 6 counts the following compositions (empty column indicated by dot):
  .  (15)  (24)    (33)      (42)     (51)   (6)
           (141)   (132)     (123)    (114)
           (1113)  (231)     (222)    (213)
           (1212)  (1122)    (321)    (312)
           (1311)  (1221)    (1131)   (411)
                   (2112)    (2121)
                   (2211)    (3111)
                   (11121)   (11112)
                   (12111)   (11211)
                   (111111)  (21111)
The reverse-alternating version is the same. Counting compositions by all three parameters (sum, length, alternating sum) gives A345197. Compositions of 2n with alternating sum 2k with k ranging from -n + 1 to n are A034871. (End)
Also the convolution triangle of A000012. - Peter Luschny, Oct 07 2022
From Sergey Kitaev, Nov 18 2023: (Start)
Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with k right-to-left maxima. A right-to-left maximum in a permutation a(1)a(2)...a(n) is position i such that a(j) < a(i) for all i < j.
Number of permutations of length n avoiding simultaneously the patterns 231 and 312 with k right-to-left minima (resp., left-to-right maxima). A right-to-left minimum (resp., left-to-right maximum) in a permutation a(1)a(2)...a(n) is position i such that a(j) > a(i) for all j > i (resp., a(j) < a(i) for all j < i).
Number of permutations of length n avoiding simultaneously the patterns 213 and 312 with k right-to-left maxima (resp., left-to-right maxima).
Number of permutations of length n avoiding simultaneously the patterns 213 and 231 with k right-to-left maxima (resp., right-to-left minima). (End)

			G.f. = 1 + x * (x + x^3 * (1 + x) + x^6 * (1 + x)^2 + x^10 * (1 + x)^3 + ...). - _Michael Somos_, Aug 20 2006
The triangle T(n, k) begins:
n\k  0 1 2  3  4   5   6  7  8 9 10 ...
0:   1
1:   0 1
2:   0 1 1
3:   0 1 2  1
4:   0 1 3  3  1
5:   0 1 4  6  4   1
6:   0 1 5 10 10   5   1
7:   0 1 6 15 20  15   6  1
8:   0 1 7 21 35  35  21  7  1
9:   0 1 8 28 56  70  56 28  8 1
10:  0 1 9 36 84 126 126 84 36 9  1
... reformatted _Wolfdieter Lang_, Jul 31 2017
From _Paul Weisenhorn_, Feb 09 2011: (Start)
T(r=5,c=3) = binomial(4,2) = 6 unordered partitions of the number n = r*(r+1)/2+c = 18 with (r+1)=6 summands: (5+5+4+2+1+1), (5+5+3+3+1+1), (5+4+4+3+1+1), (5+5+3+2+2+1), (5+4+4+2+2+1), (5+4+3+3+2+1).
T(r=5,c=3) = binomial(4,2) = 6 unordered partitions of the number n = r*(r+1)/2+(c-1) = 17 with r=5 summands: (5+5+4+2+1), (5+5+3+3+1), (5+5+3+2+2), (5+4+4+3+1), (5+4+4+2+2), (5+4+3+3+2).  (End)
From _James East_, Apr 11 2014: (Start)
a(0,0)=1 since there is a unique (order-preserving) function {}->{}.
a(m,0)=0 for m>0 since there is no function from a nonempty set to the empty set.
a(3,2)=4 because there are four order-preserving functions {1,2,3}->{1,2}: these are [1,1,1], [2,2,2], [1,1,2], [1,2,2]. Here f=[a,b,c] denotes the function defined by f(1)=a, f(2)=b, f(3)=c.
a(2,3)=6 because there are six order-preserving functions {1,2}->{1,2,3}: these are [1,1], [1,2], [1,3], [2,2], [2,3], [3,3].
(End)

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Part 1, Section 7.2.1.3, 2011.

Alois P. Heinz, Rows n = 0..140, flattened
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
James East and Peter J. McNamara, On the work performed by a transformation semigroup, Australas. J. Combin. 49 (2011), 95-109.
Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 8.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Case m=0 of the polynomials defined in A278073.
Cf. A000012 (diagonal), A011782 (row sums), A088218 (central terms).
Cf. A007318, A048004, A127952, A118800, A200139, A130595, A167374.
The terms just left of center in odd-indexed rows are A001791, even A002054.
The odd-indexed rows are A034871.
Row sums without the center are A058622.
The unordered version is A072233, without zeros A008284.
Right half without center has row sums A027306(n-1).
Right half with center has row sums A116406(n).
Left half without center has row sums A294175(n-1).
Left half with center has row sums A058622(n-1).
A025047 counts alternating compositions.
A098124 counts balanced compositions, unordered A047993.
A106356 counts compositions by number of maximal anti-runs.
A344651 counts partitions by sum and alternating sum.
A345197 counts compositions by sum, length, and alternating sum.
Cf. A000346, A000984, A001700, A008549, A008965, A114121, A124754, A238279, A345907, A345908, A346632.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      expand(add(b(n-i*j, i-1, p+j)/j!*x^j, j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, May 25 2014
# Alternatively:
T := proc(k,n) option remember;
if k=n then 1 elif k=0 then 0 else
add(T(k-1,n-i), i=1..n-k+1) fi end:
A097805 := (n,k) -> T(k,n):
for n from 0 to 12 do seq(A097805(n,k), k=0..n) od; # Peter Luschny, Mar 12 2016
# Uses function PMatrix from A357368.
PMatrix(10, n -> 1);  # Peter Luschny, Oct 07 2022

T[0, 0] = 1; T[n_, k_] := Binomial[n-1, k-1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 03 2014, after Paul Weisenhorn *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jan 23 2022 *)

{a(n) = my(m); if( n<2, n==0, n--; m = (sqrtint(8*n + 1) - 1)\2; binomial(m-1, n - m*(m + 1)/2))}; /* Michael Somos, Aug 20 2006 */

T(n,k) = if (k==0, 0^n, binomial(n-1, k-1)); \\ Michel Marcus, May 06 2022

row(n) = vector(n+1, k, k--; if (k==0, 0^n, binomial(n-1, k-1))); \\ Michel Marcus, May 06 2022

from math import comb
def T(n, k): return comb(n-1, k-1) if k != 0 else k**n  # Peter Luschny, May 06 2022

# Illustrates a basic partition formula, is not efficient as a program for large n.
def A097805_row(n):
    r = []
    for k in (0..n):
        s = 0
        for q in Partitions(n, max_part=k, inner=[k]):
            s += mul(binomial(q[j],q[j+1]) for j in range(len(q)-1))
        r.append(s)
    return r
[A097805_row(n) for n in (0..9)] # Peter Luschny, Jul 13 2015

Number triangle T(n, k) defined by T(n,k) = Sum_{j=0..n} binomial(n, j)*if(k<=j, (-1)^(j-k), 0).
T(r,c) = binomial(r-1,c-1), 0 <= c <= r. - Paul Weisenhorn, Feb 09 2011
G.f.: (-1+x)/(-1+x+x*y). - R. J. Mathar, Aug 11 2015
a(0,0) = 1, a(n,k) = binomial(n-1,n-k) = binomial(n-1,k-1) Juergen Will, Jan 04 2016
G.f.: (x^1 + x^2 + x^3 + ...)^k = (x/(1-x))^k. - Juergen Will, Jan 04 2016
From Tom Copeland, Nov 15 2016: (Start)
E.g.f.: 1 + x*[e^((x+1)t)-1]/(x+1).
This padded Pascal matrix with the odd columns negated is NpdP = M*S = S^(-1)*M^(-1) = S^(-1)*M, where M(n,k) = (-1)^n A130595(n,k), the inverse Pascal matrix with the odd rows negated, S is the summation matrix A000012, the lower triangular matrix with all elements unity, and S^(-1) = A167374, a finite difference matrix. NpdP is self-inverse, i.e., (M*S)^2 = the identity matrix, and has the e.g.f. 1 - x*[e^((1-x)t)-1]/(1-x).
M = NpdP*S^(-1) follows from the well-known recursion property of the Pascal matrix, implying NpdP = M*S.
The self-inverse property of -NpdP is implied by the self-inverse relation of its embedded signed Pascal submatrix M (cf. A130595). Also see A118800 for another proof.
Let P^(-1) be A130595, the inverse Pascal matrix. Then T = A200139*P^(-1) and T^(-1) = padded P^(-1) = P*A097808*P^(-1). (End)
The center (n>0) is T(2n+1,n+1) = A000984(n) = 2*A001700(n-1) = 2*A088218(n) = A126869(2n) = 2*A138364(2n-1). - Gus Wiseman, Jan 25 2022

Corrected by Philippe Deléham, Oct 05 2005

New name using classical terminology by Peter Luschny, Feb 05 2019

A270995 Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 23, 37, 64, 108, 180, 290, 488, 772, 1251, 2001, 3180, 4982, 7913, 12261, 19162, 29669, 45804, 70187, 108029, 164276, 250267, 379439, 574067, 864044, 1302169, 1949050, 2917900, 4352796, 6481627, 9620256, 14274080, 21090608, 31142909

Offset: 0

Vaclav Kotesovec, Mar 28 2016

nonn

The number of ways a number can be partitioned into not necessarily distinct parts and then each part is partitioned into distinct parts. Also a(n) > A089259(n) for n>5. - Gus Wiseman, Apr 10 2016
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers with weakly decreasing multiplicities. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 7 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{2},{1,1}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{1},{2},{1,1}}
                                   {{2},{3},{1,1}}
                                   {{1},{2},{3},{4}}
The weakly normal non-strict version is A055887.
The non-strict version is A063834.
The weakly normal version is A304969.
(End)

			a(6)=23: {(6), (5)(1), (51), (4)(2), (42), (4)(1)(1), (41)(1), (3)(3), (3)(2)(1), (3)(21), (32)(1), (31)(2), (21)(3), (321), (3)(1)(1)(1), (31)(1)(1), (2)(2)(2), (2)(2)(1)(1), (21)(2)(1), (21)(21), (2)(1)(1)(1)(1), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1)}.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. A063834 (twice partitioned numbers), A271619, A279784, A327554, A327608.
The unordered version is A089259, non-strict A001970 (row-sums of A061260).
For compositions instead of partitions we have A304969, non-strict A055887.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by sum and length.

nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

From Vaclav Kotesovec, Mar 28 2016: (Start)
a(n) ~ c * n^2 * 2^(n/3), where
c = 436246966131366188.9451742926272200575837456478739... if mod(n,3) = 0
c = 436246966131366188.9351143199611598469443841182807... if mod(n,3) = 1
c = 436246966131366188.9322714926383227135786894927498... if mod(n,3) = 2
(End)

A081362 Expansion of q^(1/24) * eta(q) / eta(q^2) in powers of q.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18, -20, 23, -25, 26, -29, 33, -35, 37, -41, 46, -49, 52, -57, 63, -68, 72, -78, 87, -93, 98, -107, 117, -125, 133, -144, 157, -168, 178, -192, 209, -223, 236, -255, 276, -294, 312, -335, 361, -385

Offset: 0

Michael Somos, Mar 18 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
(Number of partitions of n into an even number of parts) - (number of partitions of n into an odd number of parts). [Fine]
Number 3 of the 130 identities listed in Slater 1952. - Michael Somos, Aug 20 2015

			G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 2*x^11 + ...
G.f. = 1/q - q^23 - q^71 + q^95 - q^119 + q^143 - q^167 + 2*q^191 - 2*q^215 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 425, Corollary 1, Eqs. (37)-(40).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 38, Eq. (22.14).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold; Mansour, Toufik Counting corners in partitions. Ramanujan J. 39, No. 1, 201-224 (2016).
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=0. - _N. J. A. Sloane_, Aug 31 2014
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 13.
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-o}(n).
Lucy Joan Slater, Further Identities of the Rogers-Ramanujan Type, Proc. London Math. Soc., Series 2, vol.s2-54, no.2, pp.147-167, (1952).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000700, A069910, A069911, A072233.

a[ n_] := SeriesCoefficient[ Product[1 - x^k, {k, 1, n, 2}], {x, 0, n}];
a[ n_] := With[ {t = Log[q] / (2 Pi I)}, SeriesCoefficient[ q^(1/24) DedekindEta[t] / DedekindEta[2 t], {q, 0, n}]];
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {}, {}, x^2, x], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jan 02 2015 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {}, {-1}, x, -1] 2, {x, 0, n}]; (* Michael Somos, May 11 2015 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / ( 1 - m)^2 / (16 q))^(-1/24), {q, 0, n}]]; (* Michael Somos, May 11 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^2 + A), n))};

from sage.combinat.partition import number_of_partitions_length
print([sum((-1)^k*number_of_partitions_length(n, k) for k in (0..n)) for n in (0..69)]) # Peter Luschny, Aug 03 2015

G.f.: Product_{k>0} (1 - x^(2*k-1)) = Product_{k>0} 1 / (1 + x^k) = 1 + Sum_{k>0} (-x)^k / (Product_{i=1..k} (1 - x^i)).
a(n) = A027187(n)-A027193(n). [Fine]
This is the convolution inverse of A000009 (partitions into distinct parts) - i.e. the negation of the INVERTi transform of A000009. - Franklin T. Adams-Watters, Jan 06 2006
Expansion of chi(-x) = chi(-x^2) / chi(x) = phi(-x) / f(-x) = phi(-x^2) / f(x) = psi(-x) / f(-x^4) = f(-x) / f(-x^2) = f(-x^2) / psi(x) in powers of x where chi(), psi(), phi(), f() are Ramanujan theta functions.
Expansion of chi(x) * chi(-x) = f(x) / psi(x) = f(-x) / psi(-x) in powers of x^2 where chi(), psi(), f() are Ramanujan theta functions.
Expansion of f(-x, -x^5) / psi(x^3) = phi(-x^3) / f(x, x^2) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Jun 03 2015
Given g.f. A(x), then B(q) = A(q^3)^8 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2*v + 16*u - v^2.
G.f. A(x) satisfies A(x^2) = A(x) * A(-x).
G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A000009.
Euler transform of period 2 sequence [ -1, 0, ...].
Expansion of q^(1/24) * f1(t) in powers of q = exp(Pi i t) where f1() is a Weber function.
a(n) = (-1)^n * A000700(n). a(2*n) = A069910(n). a(2*n + 1) = - A069911(n).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*24^(1/4)*n^(3/4)). - Vaclav Kotesovec, Feb 25 2015
a(n) = Sum_{k = 0..n} (-1)^k*A072233(n,k). - Peter Luschny, Aug 03 2015
G.f.: Sum_{k>=0} (-1)^k * x^k^2 / (Product_{i=1..k} (1 - x^(2*i))). - Michael Somos, Aug 20 2015
Sum_{j>=0} a(n-A000217(j)) = 0 if n not in A152749, = (-1)^k if n is in A152749, n=k*(3*k-1). [Merca, Corollary 4.1] - R. J. Mathar, Jun 18 2016
a(n) = -(1/n)*Sum_{k=1..n} A000593(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, May 28 2018
G.f.: (1 - x) * Sum_{k >= 0} (-1)^k * x^(k*(k+2)) / (Product_{i = 1..k} (1 - x^(2*i))) = (1 - x)*(1 - x^3) * Sum_{k >= 0} (-1)^k * x^(k*(k+4)) / (Product_{i = 1..k} (1 - x^(2*i)))= (1 - x)*(1 - x^3)*(1 - x^5) *
Sum_{k >= 0} (-1)^k * x^(k*(k+6)) / (Product_{i = 1..k} (1 - x^(2*i))) = .... - Peter Bala, Jan 15 2021

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A238312 Square row sums of the table A072233 (A008284).

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A238313 Alternating square row sums of the table A072233 (A008284).

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A238314 Binomial transform of the squared rows of the table A072233 (A008284).

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A008284 Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number of partitions of n into k positive parts, 1 <= k <= n.

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A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

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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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