A000712 -id:A000712 - cp's OEIS frontend

A002720 Number of partial permutations of an n-set; number of n X n binary matrices with at most one 1 in each row and column.

1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, 234662231, 3405357682, 53334454417, 896324308634, 16083557845279, 306827170866106, 6199668952527617, 132240988644215842, 2968971263911288999, 69974827707903049154, 1727194482044146637521, 44552237162692939114282

Offset: 0

N. J. A. Sloane

a(n) is also the total number of increasing subsequences of all permutations of [1..n] (see Lifschitz and Pittel). - N. J. A. Sloane, May 06 2012
a(n) = A000142 + A001563 + A001809 + A001810 + A001811 + A001812 + ... these sequences respectively give the number of increasing subsequences of length i for i=0,1,2,... in all permutations of [1..n]. - Geoffrey Critzer, Jan 17 2013
a(n) is also the number of matchings in the complete bipartite graph K(n,n). - Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002
a(n) is also the number of 12-avoiding signed permutations in B_n (see Simion ref).
a(n) is also the order of the symmetric inverse semigroup (monoid) I_n. - A. Umar, Sep 09 2008
EXP transform of A001048(n) = n! + (n-1)!. - Franklin T. Adams-Watters, Dec 28 2006
From Peter Luschny, Mar 27 2011: (Start)
Let B_{n}(x) = Sum_{j>=0} exp(j!/(j-n)!*x-1)/j!; then a(n) = 2! [x^2] Taylor(B_{n}(x)), where [x^2] denotes the coefficient of x^2 in the Taylor series for B_{n}(x).
a(n) is column 2 of the square array representation of A090210. (End)
a(n) is the Hosoya index of the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 09 2011
a(n) is also number of non-attacking placements of k rooks on an n X n board, summed over all k >= 0. - Vaclav Kotesovec, Aug 28 2012
Also the number of vertex covers and independent vertex sets in the n X n rook graph. - Eric W. Weisstein, Jan 04 2013
a(n) is the number of injective functions from subsets of [n] to [n] where [n]={1,2,...,n}. For a subset D of size k, there are n!/(n-k)! injective functions from D to [n]. Summing over all subsets, we obtain a(n) = Sum_{k=0..n} C(n,k)*n!/(n-k)! = Sum_{k=0..n} k!*C(n,k)^2. - Dennis P. Walsh, Nov 16 2015
Also the number of cliques in the n X n rook complement graph. - Eric W. Weisstein, Sep 14 2017
a(n)/n! is the expected value of the n-th term of Ulam's "history-dependent random sequence". See Kac (1989), Eq.(2). - N. J. A. Sloane, Nov 16 2019
a(2*n) is odd and a(2*n+1) is even for all n. More generally, for each positive integer k, a(n+k) == a(n) (mod k) for all n. It follows that for each positive integer k, the sequence obtained by reducing a(n) modulo k is periodic, with period dividing k. Various divisibility properties of the sequence follow from this: for example, a(7*n+2) == 0 (mod 7), a(11*n+4) == 0 (mod 11), a(17*n+3) == 0 (mod 17) and a(19*n+4) == 0 (mod 19). - Peter Bala, Nov 07 2022
Conjecture: a(n)*k is the sum of the largest parts in all integer partitions containing their own first differences with n + 1 parts and least part k. - John Tyler Rascoe, Feb 28 2024

			G.f. = 1 + 2*x + 7*x^2 + 34*x^3 + 209*x^4 + 1546*x^5 + 13327*x^6 + 130922*x^7 + ... - _Michael Somos_, Jul 31 2018

J. M. Howie, Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995). [From A. Umar, Sep 09 2008]
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
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).
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.

Alois P. Heinz, Table of n, a(n) for n = 0..443 (first 101 terms from T. D. Noe)
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Ramified inverse and planar monoids, arXiv:2210.17461 [math.RT], 2022.
A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT], 2009.
T. Banica, The algebraic structure of quantum partial isometries, arXiv:1411.0577 [math.OA], 2014-2015.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy, and D. Gouyou-Beauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Teo Banica, Algebraic invariants of truncated Fourier matrices, arXiv preprint arXiv:1401.5023 [math.QA], 2014.
D. Borwein, S. Rankin, and L. Renner, Enumeration of injective partial transformations, Discrete Math. (1989), 73: 291-296. [From A. Umar, Sep 09 2008]
D. Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.
Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 14.
Fabio Deelan Cunden, Jakub Czartowski, Giovanni Gramegna, and A. de Oliveira Junior, Relative volume of comparable pairs under semigroup majorization, arXiv:2410.23196 [math-ph], 2024. See pp. 4, 15.
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 598.
Olof Giselsson, The Universal C*-Algebra of the Quantum Matrix Ball and its Irreducible *-Representations, arXiv:1801.10608 [math.QA], 2018.
J. Godbout, Combinatorial Properties of the Mirabolic RSK Algorithm, Thesis presented to The Faculty of the Graduate College of The University of Vermont, May 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 64.
Z. Janelidze, F. van Niekerk, and J. Viljoen, What is the maximal connected partial symmetry index of a connected graph of a given size?, arXiv:2502.00704 [math.CO], 2025. See p. 3.
Mark Kac, A history-dependent random sequence defined by Ulam, Advances in Applied Mathematics 10.3 (1989): 270-277.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 219.
Vaclav Kotesovec, Too many errors around coefficient C1 in asymptotic of sequence A002720, Sep 28 2012. [The bug in program Mathematica was fixed in version 10.2.0.0 / July 2015. _Vaclav Kotesovec_, Jul 25 2015]
V. Lifschitz and P. Pittel, The number of increasing subsequences of the random permutation J. Combin. Theory Ser. A 31 (1981), no. 1, 1--20. MR0626437 (84e:05012)
Mathematica Stack Exchange, Wrong Limit with LaguerreL, May 22 2015
W. D. Munn, The characters of the symmetric inverse semigroup, Proc. Cambridge Philos. Soc. 53 (1957), 13-18. [From A. Umar, Sep 09 2008]
K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp, and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009).
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
John Riordan, Letter, Apr 28 1976.
John Riordan, Letter to N. J. A. Sloane, Oct 01 1978.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles. (Annotated scans of some selected pages)
R. Simion, Combinatorial statistics on type-B analogues of non-crossing partitions and restricted permutations, Electronic J. of Comb. 7 (2000), Art #R9.
A. Umar, Some combinatorial problems in the theory of symmetric ..., Algebra Disc. Math. 9 (2010) 115-126.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
D. P. Walsh, Notes on functions from subsets of {1,2,...,n} into {1,2,...,n}.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Matching.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Eric Weisstein's World of Mathematics, Rook Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for sequences related to Laguerre polynomials

Main diagonal of A088699. Column of A283500. Row sums of A144084.
Cf. A000110, A020556, A069223, A000712, A001048, A090210, A002793, A073003, A000262.
Column k=1 of A289192.
Cf. A160617, A160618.
Cf. A364673.

[Factorial(n)*Evaluate(LaguerrePolynomial(n), -1): n in [0..25]]; // G. C. Greubel, Aug 11 2022

A002720 := proc(n) exp(-x)*n!*hypergeom([n+1], [1], x); simplify(subs(x=1, %)) end: seq(A002720(n), n=0..25); # Peter Luschny, Mar 30 2011
A002720 := proc(n)
    option remember;
    if n <= 1 then
        n+1 ;
    else
        2*n*procname(n-1)-(n-1)^2*procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Mar 09 2017

Table[n! LaguerreL[n, -1], {n, 0, 25}]
Table[(-1)^n*HypergeometricU[-n, 1, -1], {n, 0, 25}] (* Jean-François Alcover, Jul 15 2015 *)
RecurrenceTable[{(n+1)^2 a[n] - 2(n+2) a[n+1] + a[n+2]==0, a[1]==2, a[2]==7}, a, {n, 25}] (* Eric W. Weisstein, Sep 27 2017 *)

a(n) = sum(k=0, n, k!*binomial(n, k)^2 );

a(n) = suminf ( k=0, binomial(n+k,n)/k! ) / ( exp(1)/n! ) /* Gottfried Helms, Nov 25 2006 */

{a(n)=n!^2*polcoeff(exp(x+x*O(x^n))*sum(m=0,n,x^m/m!^2),n)} /* Paul D. Hanna, Nov 18 2011 */

{a(n)=if(n==0,1,polcoeff(1-sum(m=0, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^2), n))} /* Paul D. Hanna, Nov 27 2012 */

my(x='x+O('x^22)); Vec(serlaplace((1/(1-x))*exp(x/(1-x)))) \\ Joerg Arndt, Aug 11 2022

from math import factorial, comb
def A002720(n): return sum(factorial(k)*comb(n,k)**2 for k in range(n+1)) # Chai Wah Wu, Aug 31 2023

[factorial(n)*laguerre(n, -1) for n in (0..25)] # G. C. Greubel, Aug 11 2022

a(n) = Sum_{k=0..n} k!*C(n, k)^2.
E.g.f.: (1/(1-x))*exp(x/(1-x)). - Don Knuth, Jul 1995
D-finite with recurrence: a(n) = 2*n*a(n-1) - (n-1)^2*a(n-2).
a(n) = Sum_{k>=0} (k+n)! / ((k!)^2*exp(1)). - Robert G. Wilson v, May 02 2002 [corrected by Vaclav Kotesovec, Aug 28 2012]
a(n) = Sum_{m>=0} (-1)^m*A021009(n, m). - Philippe Deléham, Mar 10 2004
a(n) = Sum_{k=0..n} C(n, k)n!/k!. - Paul Barry, May 07 2004
a(n) = Sum_{k=0..n} P(n, k)*C(n, k); a(n) = Sum_{k=0..n} n!^2/(k!*(n-k)!^2). - Ross La Haye, Sep 20 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*Bell(k+1). - Vladeta Jovovic, Mar 18 2005
Define b(n) by b(0) = 1, b(n) = b(n-1) + (1/n) * Sum_{k=0..n-1} b(k). Then b(n) = a(n)/n!. - Franklin T. Adams-Watters, Sep 05 2005
Asymptotically, a(n)/n! ~ (1/2)*Pi^(-1/2)*exp(-1/2 + 2*n^(1/2))/n^(1/4) and so a(n) ~ C*BesselI(0, 2*sqrt(n))*n! with C = exp(-1/2) = 0.6065306597126334236... - Alec Mihailovs, Sep 06 2005, establishing a conjecture of Franklin T. Adams-Watters
a(n) = (n!/e) * Sum_{k>=0} binomial(n+k,n)/k!. - Gottfried Helms, Nov 25 2006
Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem): a(n) = Integral_{x=0..oo} x^n*BesselI(0,2*sqrt(x))*exp(-x)/exp(1) dx, n >= 0. - Karol A. Penson and G. H. E. Duchamp (gduchamp2(AT)free.fr), Jan 09 2007
a(n) = n! * LaguerreL[n, -1].
E.g.f.: exp(x) * Sum_{n>=0} x^n/n!^2 = Sum_{n>=0} a(n)*x^n/n!^2. - Paul D. Hanna, Nov 18 2011
From Peter Bala, Oct 11 2012: (Start)
Denominators in the sequence of convergents coming from Stieltjes's continued fraction for A073003, the Euler-Gompertz constant G := Integral_{x = 0..oo} 1/(1+x)*exp(-x) dx:
G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The numerators are in A002793. (End)
G.f.: 1 = Sum_{n>=0} a(n) * x^n * (1 - (n+1)*x)^2. - Paul D. Hanna, Nov 27 2012
E.g.f.: exp(x/(1-x))/(1-x) = G(0)/(1-x) where G(k) = 1 + x/((2*k+1)*(1-x) - x*(1-x)*(2*k+1)/(x + (1-x)*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 28 2012
a(n) = Sum_{k=0..n} L(n,k)*(k+1); L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014
a(n) = n! * A160617(n)/A160618(n). - Alois P. Heinz, Jun 28 2017
0 = a(n)*(-24*a(n+2) +99*a(n+3) -78*a(n+4) +17*a(n+5) -a(n+6)) +a(n+1)*(-15*a(n+2) +84*a(n+3) -51*a(n+4) +6*a(n+5)) +a(n+2)*(-6*a(n+2) +34*a(n+3) -15*a(n+4)) +a(n+3)*(+10*a(n+3)) for all n>=0. - Michael Somos, Jul 31 2018
a(n) = Sum_{k=0..n} C(n,k)*k!*A000262(n-k). - Geoffrey Critzer, Jan 07 2023
a(n) = A000262(n+1) - n * A000262(n). - Werner Schulte, Mar 29 2024
a(n) = denominator of (1 + n/(1 + n/(1 + (n-1)/(1 + (n-1)/(1 + ... + 1/(1 + 1/(1))))))). See A000262 for the numerators. - Peter Bala, Feb 11 2025

2nd description from R. H. Hardin, Nov 1997

3rd description from Wouter Meeussen, Jun 01 1998

A103919 Triangle of numbers of partitions of n with total number of odd parts equal to k from {0,...,n}.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1, 0, 4, 0, 2, 0, 1, 3, 0, 5, 0, 2, 0, 1, 0, 7, 0, 5, 0, 2, 0, 1, 5, 0, 9, 0, 5, 0, 2, 0, 1, 0, 12, 0, 10, 0, 5, 0, 2, 0, 1, 7, 0, 17, 0, 10, 0, 5, 0, 2, 0, 1, 0, 19, 0, 19, 0, 10, 0, 5, 0, 2, 0, 1, 11, 0, 28, 0, 20, 0, 10, 0, 5, 0, 2, 0, 1, 0, 30, 0, 33, 0, 20, 0, 10, 0, 5, 0, 2, 0, 1

Offset: 0

Wolfdieter Lang, Mar 24 2005

The partition (0) of n=0 is included. For n>0 no part 0 appears.
The first (k=0) column gives the number of partitions without odd parts, i.e., those with even parts only. See A035363.
Without the alternating zeros this becomes a triangle with columns given by the rows of the S_n(m) table shown in the Riordan reference.
From Gregory L. Simay, Oct 31 2015: (Start)
T(2n+k,k) = the number of partitions of n with parts 1..k of two kinds. If n<=k, then T(2n+k) = A000712(n), the number of partitions of n with parts of two kinds.
T(2n+k) = the convolution of A000041(n) and the number of partitions of n+k having exactly k parts.
T(2n+k) = d(n,k) where d(n,0) = p(n) and d(n,k) = d(n,k-1) + d(n-k,k-1) + d(n-2k,k-1) + ... (End)
From Emeric Deutsch, Oct 04 2016: (Start)
T(n,k) = number of partitions (p1 >= p2 >= p3 >= ...) of n having alternating sum p1 - p2 + p3 - ... = k. Example: T(5,3) = 2 because there are two partitions (3,1,1) and (4,1) of 5 with alternating sum 3.
The equidistribution of the partition statistics "alternating sum" and "total number of odd parts" follows by conjugation. (End)

			The triangle a(n,k) begins:
n\k 0  1  2  3  4  5  6  7  8  9 10
0:  1
1:  0  1
2:  1  0  1
3:  0  2  0  1
4:  2  0  2  0  1
5:  0  4  0  2  0  1
6:  3  0  5  0  2  0  1
7:  0  7  0  5  0  2  0  1
8:  5  0  9  0  5  0  2  0  1
9:  0 12  0 10  0  5  0  2  0  1
10: 7  0 17  0 10  0  5  0  2  0  1
... Reformatted - _Wolfdieter Lang_, Apr 28 2013
a(0,0) = 1 because n=0 has no odd part, only one even part, 0, by definition. a(5,3) = 2 because there are two partitions (1,1,3) and (1,1,1,2) of 5 with exactly 3 odd parts.
From _Gregory L. Simay_, Oct 31 2015: (Start)
T(10,4) = T(2*3+4,4) = d(3,4) = A000712(3) = 10.
T(10,2) = T(2*4+2,2) = d(4,2) = d(4,1)+d(2,1)+d(0,1) = d(4,0)+d(3,0)+d(2,0)+d(1,0)+d(0,0) + d(2,0)+d(1,0)+d(0,0) + d(0,0) = convolution sum p(4)+p(3)+2*p(2)+2*p(1)+3*p(0) = 5+3+2*2+2*1+3*1 = 17.
T(9,1) = T(8,0) + T(7,1) = 5 + 7 = 12.
(End)

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Alois P. Heinz, Rows n = 0..140, flattened
D. Kim, A. J. Yee, A note on partitions into distinct parts and odd parts, Ramanujan J. 3 (1999), 227-231. [_R. J. Mathar_, Nov 11 2008]
Wolfdieter Lang, First 11 rows.

Row sums gives A000041 (partition numbers). Columns: k=0: A035363 (with zero entries) A000041 (without zero entries), k=1: A000070, k=2: A000097, k=3: A000098, k=4: A000710, 3k>=n: A000712.
Cf. A066897.
The strict version (without zeros) is A152146 interleaved with A152157.
The rows (without zeros) are A239830 interleaved with A239829.
The reverse version (without zeros) is the right half of A344612.
Removing all zeros gives A344651.
The strict reverse version (without zeros) is the right half of A344739.
Cf. A006330, A027187, A116406, A344607, A344608, A344649, A344654.

g:=1/product((1-t*x^(2*j-1))*(1-x^(2*j)),j=1..20): gser:=simplify(series(g,x=0,22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser,x^n) od: for n from 0 to 18 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form # Emeric Deutsch, Feb 17 2006

T[n_, k_] := T[n, k] = Which[nJean-François Alcover, Mar 05 2014, after Paul D. Hanna *)
Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]==k&]],{n,0,15},{k,0,n}] (* _Gus Wiseman, Jun 20 2021 *)

{T(n, k)=if(n>=k, if(n==k, 1, if((n-k+1)%2==0, 0, if(k==0, sum(m=0, n, T(n\2, m)), T(n-1, k-1)+T(n-2*k, k)))))}
for(n=0, 20, for(k=0, n, print1(T(n, k), ", ")); print(""))
\\ Paul D. Hanna, Apr 27 2013

a(n, k) = number of partitions of n>=0, which have exactly k odd parts (and possibly even parts) for k from {0, ..., n}.
Sum_{k=0..n} k*T(n,k) = A066897(n). - Emeric Deutsch, Feb 17 2006
G.f.: G(t,x) = 1/Product_{j>=1} (1-t*x^(2*j-1))*(1-x^(2*j)). - Emeric Deutsch, Feb 17 2006
G.f. T(2n+k,k) = g.f. d(n,k) = (1/Product_{j=1..k} (1-x^j)) * g.f. p(n). - Gregory L. Simay, Oct 31 2015
T(n,k) = T(n-1,k-1) + T(n-2k,k). - Gregory L. Simay, Nov 01 2015

A001523 Number of stacks, or planar partitions of n; also weakly unimodal compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 27, 47, 79, 130, 209, 330, 512, 784, 1183, 1765, 2604, 3804, 5504, 7898, 11240, 15880, 22277, 31048, 43003, 59220, 81098, 110484, 149769, 202070, 271404, 362974, 483439, 641368, 847681, 1116325, 1464999, 1916184, 2498258, 3247088, 4207764

Offset: 0

N. J. A. Sloane

a(n) counts stacks of integer-length boards of total length n where no board overhangs the board underneath.
Number of graphical partitions on 2n nodes that contain a 1. E.g. a(3)=4 and so there are 4 graphical partitions of 6 that contain a 1, namely (111111), (21111), (2211) and (3111). Only (222) fails. - Jon Perry, Jul 25 2003
It would seem from Stanley that he regards a(0)=0 for this sequence and A001522. - Michael Somos, Feb 22 2015
In the article by Auluck is a typo in the formula (24), 2*Pi is missing in an exponent on the left side of the equation for Q(n). The correct term is exp(2*Pi*sqrt(n/3)), not just exp(sqrt(n/3)). - Vaclav Kotesovec, Jun 22 2015

			For a(4)=8 we have the following stacks:
x
x x. .x
x x. .x x.. .x. ..x xx
x xx xx xxx xxx xxx xx xxxx
G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 27*x^6 + 47*x^7 + 79*x^8 + ...
From _Gus Wiseman_, Mar 04 2020: (Start)
The a(1) = 1 through a(5) = 15 unimodal compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (112)   (41)
                    (121)   (113)
                    (211)   (122)
                    (1111)  (131)
                            (221)
                            (311)
                            (1112)
                            (1121)
                            (1211)
                            (2111)
                            (11111)
(End)

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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686, g(x).
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy)
H. Bottomley, Illustration of initial terms
Shouvik Datta, Matthias R. Gaberdiel, Wei Li, and Cheng Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, U(1,z).
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 46.
Rigoberto Flórez, José L. Ramírez, and Diego Villamizar, Restricted bargraphs and unimodal compositions, J. Comb. Theory, Series A, (2024) Vol. 208, Art. No. 105934.
R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms
Alan D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003 [math.CO], 2011.
Eric Weisstein's World of Mathematics, Unimodal Sequence
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A054250, A059618, A059623, A001522, A001524.
Cf. A000569. Bisections give A100505, A100506.
Row sums of A247255.
Row sums of A072704.
The strict case is A072706.
The complement is counted by A115981.
The case covering an initial interval is A227038.
The version whose negation is unimodal as well appears to be A329398.
Unimodal sequences covering an initial interval are A007052.
Non-unimodal permutations are A059204.
Non-unimodal sequences covering an initial interval are A328509.
Partitions with unimodal run-lengths are A332280.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
The number of unimodal permutations of the prime indices of n is A332288.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Cf. A107429, A156253, A332285, A332294, A332577, A332642, A332669.

m:=100;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1 + (&+[ x^n*(1-x^n)/(&*[(1-x^j)^2: j in [1..n]]): n in [1..m+2]]) )); // G. C. Greubel, Apr 03 2023

b:= proc(n, i) option remember;
      `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+
      add(b(n-i*j, i+1)*(j+1), j=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014

max = 40; s = 1 + Sum[(-1)^(k + 1)*q^(k*(k + 1)/2), {k, 1, max}] / QPochhammer[q]^2 + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Jan 25 2012, updated Nov 29 2015 *)
b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i]==0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 0, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 24 2015, after Alois P. Heinz *)
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],unimodQ[#]&]],{n,0,10}] (* Gus Wiseman, Mar 04 2020 *)

{a(n) = if( n<1, n==0, polcoeff( sum(k=1, (sqrt(1 + 8*n) - 1)\2, -(-1)^k * x^((k + k^2)/2)) / eta(x + x * O(x^n))^2 ,n))}; /* Michael Somos, Jul 22 2003 */

def b(n, i):
    if i>n: return 0
    if n%i==0: x=1
    else: x=0
    return x + sum([b(n - i*j, i + 1)*(j + 1) for j in range(n//i + 1)])
def a(n): return 1 if n==0 else b(n, 1) # Indranil Ghosh, Jun 09 2017, after Maple code by Alois P. Heinz

a(n) = Sum_{k=1..n} f(k, n-k), where f(n, k) (= A054250) = 1 if k = 0; Sum_{j=1..min(n, k)} (n-j+1)*f(j, k-j) if k > 0. - David W. Wilson, May 05 2000
a(n) = Sum_{k} A059623(n, k) for n > 0. - Henry Bottomley, Feb 01 2001
A006330(n) + a(n) = A000712(n). - Michael Somos, Jul 22 2003
G.f.: 1 + (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k))^2. - Michael Somos, Jul 22 2003
G.f.: 1 + Sum_{n>=1} (x^n / ( ( Product_{k=1..n-1} (1 - x^k)^2 ) * (1-x^n) ) ). - Joerg Arndt, Oct 01 2012
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015
a(n) + A115981(n) = 2^(n - 1). - Gus Wiseman, Mar 04 2020

More terms from David W. Wilson, May 05 2000

Definition corrected by Wolfdieter Lang, Dec 05 2018

A115994 Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 2, 6, 5, 7, 8, 8, 14, 9, 20, 1, 10, 30, 2, 11, 40, 5, 12, 55, 10, 13, 70, 18, 14, 91, 30, 15, 112, 49, 16, 140, 74, 1, 17, 168, 110, 2, 18, 204, 158, 5, 19, 240, 221, 10, 20, 285, 302, 20, 21, 330, 407, 34, 22, 385, 536, 59, 23, 440, 698, 94, 24, 506, 896, 149, 25

Offset: 1

Emeric Deutsch, Feb 11 2006

Row n has floor(sqrt(n)) terms. Row sums yield A000041. Column 2 yields A006918. sum(k*T(n,k),k=1..floor(sqrt(n)))=A115995.
T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.
The limit of the diagonals is A000712 (partitions into parts of two kinds). In particular, if 0<=m<=n, T(n(n+1)/2 + m, n) = A000712(m). These partitions in this range can be viewed as an equilateral right triangle of side n, with one partition appended on the top (at the left) and another appended on the right. - Franklin T. Adams-Watters, Jan 11 2006
Successive columns approach closer and closer to A000712. - N. J. A. Sloane, Mar 10 2007

			T(5,2) = 2 because the only partitions of 5 having Durfee square of size 2 are [3,2] and [2,2,1]; the other five partitions ([5], [4,1], [3,1,1], [2,1,1,1] and [1,1,1,1,1]) have Durfee square of size 1.
Triangle starts:
  1;
  2;
  3;
  4,  1;
  5,  2;
  6,  5;
  7,  8;
  8, 14;
  9, 20,  1;
  ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Alois P. Heinz, Rows n = 1..1000, flattened
E. R. Canfield, From recursions to asymptotics: Durfee and dilogarithmic deductions, Advances in Applied Mathematics, Volume 34, Issue 4, May 2005, Pages 768-797
E. R. Canfield, S. Corteel, C. D. Savage, Durfee Polynomials, Electronic Journal of Combinatorics 5 (1998), #R32.
S. B. Ekhad, D. Zeilberger, A Quick Empirical Reproof of the Asymptotic Normality of the Hirsch Citation Index (First proved by Canfield, Corteel, and Savage), arXiv preprint arXiv:1411.0002, 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 45
Eric Weisstein's World of Mathematics, Durfee Square.

For another version see A115720. Row lengths A000196.

Cf. A115995, A115721, A115722, A008284, A006918.

g:=sum(t^k*q^(k^2)/product((1-q^j)^2,j=1..k),k=1..40): gser:=series(g,q=0,32): for n from 1 to 27 do P[n]:=coeff(gser,q^n) od: for n from 1 to 27 do seq(coeff(P[n],t^j),j=1..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2):
seq(seq(T(n, k), k=1..floor(sqrt(n))), n=1..30); # Alois P. Heinz, Apr 09 2012

Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[ Series[x^(n^2)/Product[1-x^i,{i,1,n}]^2,{x,0,nn}],x],{n,1,10}]],1]] //Grid (* Geoffrey Critzer, Sep 27 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := Sum[b[m, k]*b[n-k^2-m, k], {m, 0, n-k^2}]; Table[T[n, k], {n, 1, 30}, {k, 1, Sqrt[n]}] // Flatten (* Jean-François Alcover, Dec 25 2015, after Alois P. Heinz *)

G.f.: sum(k>=1, t^k*q^(k^2)/product(j=1..k, (1-q^j)^2 ) ).

T(n,k) = Sum_{i=0}^{n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.

Edited and verified by Franklin T. Adams-Watters, Mar 11 2006

A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

Original entry on oeis.org

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 65, 69, 74, 77, 84, 86, 90, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 156, 158, 161, 177, 178, 185, 196, 198, 201, 202, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 278, 287, 291, 299

Offset: 1

Gus Wiseman, May 17 2019

nonn

These are Heinz numbers of the integer partitions counted by A045931.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The integers in the multiplicative subgroup of positive rational numbers generated by the products of two consecutive primes (A006094). The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 6. For example, A003961(6) = 15, A003961(15) = 35, 6 * 35 = 210, 210/15 = 14. Closed also under A297845, since A297845 can be defined using squaring, prime shift and multiplication. - Peter Munn, Oct 05 2020

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   14: {1,4}
   15: {2,3}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   51: {2,7}
   58: {1,10}
   65: {3,6}
   69: {2,9}
   74: {1,12}
   77: {4,5}
   84: {1,1,2,4}
   86: {1,14}
   90: {1,2,2,3}
   93: {2,11}
   95: {3,8}

David A. Corneth, Table of n, a(n) for n = 1..10000

Positions of 0's in A195017.
A257992(n) = A257991(n).
Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
Closed under: A003961, A003991, A297845.
Subsequence of A028260, A332820.

Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k*(-1)^PrimePi[p]]]==0&]

is(n) = {my(v = vector(2), f = factor(n));for(i = 1, #f~,v[1 + primepi(f[i, 1])%2]+=f[i, 2]);v[1] == v[2]} \\ David A. Corneth, Oct 06 2020

from sympy import factorint, primepi
def ok(n):
    v = [0, 0]
    for p, e in factorint(n).items(): v[primepi(p)%2] += e
    return v[0] == v[1]
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, Apr 16 2022 after David A. Corneth

A045931 Number of partitions of n with equal number of even and odd parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 5, 7, 9, 11, 16, 18, 25, 28, 41, 44, 62, 70, 94, 107, 140, 163, 207, 245, 302, 361, 440, 527, 632, 763, 904, 1090, 1285, 1544, 1812, 2173, 2539, 3031, 3538, 4202, 4896, 5793, 6736, 7934, 9221, 10811, 12549, 14661, 16994, 19780

Offset: 0

David W. Wilson

nonn

The trivariate g.f. with x marking weight (i.e., sum of the parts), t marking number of odd parts and s marking number of even parts, is 1/product((1-tx^(2j-1))(1-sx^(2j)), j=1..infinity). - Emeric Deutsch, Mar 30 2006

			a(9) = 5 because we have [8,1], [7,2], [6,3], [5,4] and [2,2,2,1,1,1].
From _Gus Wiseman_, Jan 23 2022: (Start)
The a(0) = 1 through a(12) = 9 partitions (A = 10, empty columns indicated by dots):
  ()  .  .  21   .  32   2211   43   3221   54       3322   65       4332
                    41          52   4211   63       4321   74       4431
                                61          72       4411   83       5322
                                            81       5221   92       5421
                                            222111   6211   A1       6321
                                                            322211   6411
                                                            422111   7221
                                                                     8211
                                                                     22221111
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..3500 (first 1001 terms from David W. Wilson)

The version for subsets of {1..n} is A001405.
Dominated by A027187 (partitions of even length).
More odd/even parts: A108950/A108949.
More or same number of odd/even parts: A130780/A171966.
The strict case is A239241.
This is column k = 0 of the triangle A240009.
Counting only distinct parts gives A241638, ranked by A325700.
A half-conjugate version is A277579.
These partitions are ranked by A325698.
A000041 counts integer partitions, strict A000009.
A047993 counts balanced partitions, ranked by A106529.
A257991/A257992 count odd/even parts by Heinz number.
Cf. A000070, A000700, A000712, A027193, A035363, A097613, A195017.

g:=1/product((1-t*x^(2*j-1))*(1-s*x^(2*j)),j=1..30): gser:=simplify(series(g,x=0,56)): P[0]:=1: for n from 1 to 53 do P[n]:=subs(s=1/t,coeff(gser,x^n)) od: seq(coeff(t*P[n],t),n=0..53); # Emeric Deutsch, Mar 30 2006

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, ?OddQ] == Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 10}] (* partitions of n with # odd parts = # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 0, 30}] (* A045931 *)
(* Peter J. C. Moses, Mar 10 2014 *)
nmax = 100; CoefficientList[Series[Sum[x^(3*k) / Product[(1 - x^(2*j))^2, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2025 *)

G.f.: Sum_{k>=0} x^(3*k)/Product_{i=1..k} (1-x^(2*i))^2. - Vladeta Jovovic, Aug 18 2007
a(n) = A000041(n)-A171967(n) = A130780(n)-A108950(n) = A171966(n)-A108949(n). - Reinhard Zumkeller, Jan 21 2010
a(n) = A000041(n) - A108950(n) - A108949(n) = A130780(n) + A171966(n) - A000041(n). - Gus Wiseman, Jan 23 2022
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (48*n^(3/2)). - Vaclav Kotesovec, Jun 15 2025

A122768 Number of combinations which can be taken from the integer partitions of n. Total number of cases in the (n,m)-fragmentation process.

Original entry on oeis.org

0, 1, 3, 7, 15, 29, 54, 95, 163, 270, 439, 696, 1088, 1669, 2530, 3780, 5591, 8173, 11845, 17000, 24215, 34210, 48008, 66895, 92660, 127554, 174651, 237830, 322297, 434625, 583524, 779972, 1038356, 1376787, 1818755, 2393775, 3139812, 4104433, 5348375, 6947545, 8998201, 11620313, 14965126, 19220569

Offset: 0

Thomas Wieder, Sep 11 2006

nonn

Consider a fragmentation process of an n-object which consists of n unlabeled elements (= 1-parts). By definition the n-object can scatter into up to n m-parts where an m-part consists of 1 up to n elements. A 4-object can split up for example into 4 1-parts which corresponds to the integer partition [1,1,1,1], or it can, for example, rest unfragmented which corresponds to [4]. Since the number of integer partitions of n=4 equals 5, there are 5 n=4-fragmentation processes.
Now we ask for the probability of getting an m-part after an n-fragmentation. Think of a Greek statue which had been broken into n parts and covered by earth. We could find several m-parts, in the most lucky case we would find all m-parts which add up to m_1+m_2+...+m_n=n. Then the statue could be restored.
For example for n=4 we could ask for the probability prob(n=4,m=2) of just a single 2-part. We have 2 cases for a 2-part and we have 15 cases in total, thus prob(n=4,m=2)=2/15 (the 2 cases come from [1,1,2] and [2,2]). The chances to find the two 2-parts from the [2,2]-fragmentation are 1/15 only. The chances to find the n=4-object unsplitted are also 1/15 only.
This sequence is generated over the unordered partitions; for example, when n = 4 there are 1+3+2+5+4 = 15 cases. If we allow a null case for each of the five partitions then we have 15+5 = 20 which is A000712(4). - Alford Arnold, Dec 12 2006
Number of partitions into two kinds of parts with the first kind of parts used in each partition. - Joerg Arndt, Jun 21 2011

			a(n=4) = 15 because the possible combinations of all five integer partitions of n=4 are: [1], [1, 1], [1, 1, 1], [1, 1, 1, 1], [1], [2], [1, 1], [1, 2], [1, 1, 2], [2], [2, 2], [1], [3], [1, 3], [4].

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

Cf. A000041, A000079, A000262.
Cf. A000712, A074139, A074141.
Cf. A048574, A144064.

a122768 n = a122768_list !! n
a122768_list = 0 : f (tail a000041_list) [1] where
   f (p:ps) rs = (sum $ zipWith (*) rs $ tail a000041_list) : f ps (p : rs)
-- Reinhard Zumkeller, Nov 09 2015

A122768 := proc(n::integer) local i,j,prttnlst,prttn,ZahlTeile,H; prttnlst:=partition(n); H := NULL; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; ZahlTeile := nops(prttn); for j from 1 to ZahlTeile do H := H,op(choose(prttn,j)); od; od; print(n,H,nops([H])); end proc;
A000712 := proc(n) option remember ; add(combinat[numbpart](k)*combinat[numbpart](n-k),k=0..n) ; end: A000041 := proc(n) combinat[numbpart](n) ; end: A122768 := proc(n::integer) RETURN( A000712(n)-A000041(n)) ; end: for n from 0 to 80 do printf("%d,",A122768(n)) ; od: # R. J. Mathar, Aug 25 2008
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
      k*numtheory[sigma](j)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n,2)-b(n,1):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 31 2017

1/QPochhammer[x]^2 - 1/QPochhammer[x] + O[x]^50 // CoefficientList[#, x]& (* Jean-François Alcover, Feb 05 2017, after Joerg Arndt *)

x='x+O('x^66); /* that many terms */
Vec(1/eta(x)^2-1/eta(x)) /* show terms (omitting initial zero) */
/* Joerg Arndt, Jun 21 2011 */

from sympy import npartitions
def A122768(n): return (sum(npartitions(k)*npartitions(n-k) for k in range(1,n+1>>1))<<1) + (0 if n&1 else npartitions(n>>1)**2) + npartitions(n) if n else 0 # Chai Wah Wu, Sep 25 2023

G.f.: 1/P(x)^2 - 1/P(x) where P(x)=prod(k>=1, 1-x^k ). - Joerg Arndt, Jun 21 2011
With sum_i^P(n) = the sum over all P(n) integer partitions of n, sum_j^p(i) = the sum over all p(i) parts of the i-th integer partition, prttn(i) = the i-th partition whereat prttn(i) is a list, choose(L,k) = construct the list LC of combinations of a list L (see Maple), |LC| = number of elements of list LC (=Maple's nops command) we have a(n) = sum_i^P(n) sum_j^p(i) |choose(prttn,j)|
a(n) = A000712(n) - A000041(n). - Alford Arnold, Dec 12 2006
a(n) = A144064(n,2)-A144064(n,1). - Alois P. Heinz, Mar 31 2017
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)) * (1 - (Pi/12 + 45/(16*Pi))/sqrt(3*n)). - Vaclav Kotesovec, Mar 31 2017

Extended by R. J. Mathar, Aug 25 2008

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.

Original entry on oeis.org

1, 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208, 30710464, 186753920, 1171979904, 7573069568, 50305536256, 342949298688, 2396286830080, 17138748412928, 125336396368896, 936222729254912, 7136574106003456, 55466948299223040, 439216305474605056, 3540846129311916032

Offset: 0

N. J. A. Sloane

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
Also the value of the n-th derivative of exp(x^2) evaluated at 1. - N. Calkin, Apr 22 2010
For n >= 1, a(n) is also the sum of the degrees of the irreducible representations of the group of n X n signed permutation matrices (described in sequence A066051). The similar sum for the "ordinary" symmetric group S_n is in sequence A000085. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 12 2002
It appears that this is also the number of permutations of 1, 2, ..., n+1 such that each term (after the first) is within 2 of some preceding term. Verified for n+1 <= 6. E.g., a(4) = 20 because of the 24 permutations of 1, 2, 3, 4, the only ones not permitted are 1, 4, 2, 3; 1, 4, 3, 2; 4, 1, 2, 3; and 4, 1, 3, 2. - Gerry Myerson, Aug 06 2003
Hankel transform is A108400. - Paul Barry, Feb 11 2008
From Emeric Deutsch, Jun 19 2010: (Start)
Number of symmetric involutions of [2n]. Example: a(2)=6 because we have 1234, 2143, 1324, 3412, 4231, and 4321. See the Egge reference, pp. 419-420.
Number of symmetric involutions of [2n+1]. Example: a(2)=6 because we have 12345, 14325, 21354, 45312, 52341, and 54321. See the Egge reference, pp. 419-420.
(End)
Binomial convolution of sequence A000085: a(n) = Sum_{k=0..n} binomial(n,k)*A000085(k)*A000085(n-k). - Emanuele Munarini, Mar 02 2016
The sequence can be obtained from the infinite product of 2 X 2 matrices [(1,N); (1,1)] by extracting the upper left terms, where N = (1, 3, 5, ...), the odd integers. - Gary W. Adamson, Jul 28 2016
Apparently a(n) is the number of standard domino tableaux of size 2n, where a domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. - Gus Wiseman, Feb 25 2018

			G.f. = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 312*x^5 + 1384*x^6 + 6512*x^7 + ...
The a(3) = 20 domino tableaux:
1 1 2 2 3 3
.
1 2 2 3 3
1
.
1 2 3 3   1 1 3 3   1 1 2 2
1 2       2 2       3 3
.
1 1 3 3   1 1 2 2
2         3
2         3
.
1 2 3   1 2 2   1 1 3
1 2 3   1 3 3   2 2 3
.
1 3 3   1 2 2
1       1
2       3
2       3
.
1 2   1 1   1 1
1 2   2 3   2 2
3 3   2 3   3 3
.
1 3   1 2   1 1
1 3   1 2   2 2
2     3     3
2     3     3
.
1 1
2
2
3
3
.
1
1
2
2
3
3 - _Gus Wiseman_, Feb 25 2018

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.1.4 Exer. 31.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
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).

T. D. Noe, Table of n, a(n) for n=0..200
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 10.
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. 10.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
R. A. Brualdi, Shi-Mie Ma, Enumeration of involutions by descents and symmetric matrices, Eur. J. Combin. 43 (2015) 220-228
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C.-O. Chow, Counting involutory, unimodal, and alternating signed permutations, Discr. Math. 306 (2006), 2222-2228.
R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv:1302.6150 [math.RT], 2013-2014.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Yen-chi R. Lin, Asymptotic Formula for Symmetric Involutions, arXiv:1310.0988 [math.CO], 2013.
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
D. P. Roberts and A. Venkatesh, Hurwitz monodromy and full number fields, 2014.
Index entries for sequences related to Hermite polynomials

Column k=2 of A294042 and A376826.
Row sums of A122832 and A185296.
Cf. A000085, A000712, A004003, A066051, A099390, A100861, A135401, A138178, A153452, A297388, A299699, A299926, A300056, A300060.

a000898 n = a000898_list !! n
a000898_list = 1 : 2 : (map (* 2) $
   zipWith (+) (tail a000898_list) (zipWith (*) [1..] a000898_list))
-- Reinhard Zumkeller, Oct 10 2011

# For Maple program see A000903.
seq(simplify((-I)^n*HermiteH(n, I)), n=0..25); # Peter Luschny, Oct 23 2015

a[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 17 2011, after Vladeta Jovovic *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2(a[n-1]+(n-1)a[n-2])},a,{n,30}] (* Harvey P. Dale, Aug 04 2012 *)
Table[Abs[HermiteH[n, I]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
a[ n_] := Sum[ 2^(n - 2 k) n! / (k! (n - 2 k)!), {k, 0, n/2}]; (* Michael Somos, Oct 23 2015 *)

makelist((%i)^n*hermite(n,-%i),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp(2*x + x^2 + x * O(x^n)), n))}; /* Michael Somos, Feb 08 2004 */

{a(n) = if( n<2, max(0, n+1), 2*a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Feb 08 2004 */

my(x='x+O('x^66)); Vec(serlaplace(exp(2*x+x^2))) \\ Joerg Arndt, Oct 04 2013

{a(n) = sum(k=0, n\2, 2^(n - 2*k) * n! / (k! * (n - 2*k)!))}; /* Michael Somos, Oct 23 2015 */

a(n) = Sum_{m=0..n} |A060821(n,m)| = H(n,-i)*i^n, with the Hermite polynomials H(n,x); i.e., these are row sums of the unsigned triangle A060821.
E.g.f.: exp(x*(x + 2)).
a(n) = 2 * A000902(n) for n >= 1.
a(n) = Sum_{k=0..n} binomial(n,2k)*binomial(2k,k)*k!*2^(n-2k). - N. Calkin, Apr 22 2010
Binomial transform of A047974. - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k). - Vladeta Jovovic, Oct 01 2003
From Paul Barry, Aug 29 2005: (Start)
a(n) = Sum_{k=0..floor(n/2)} A001498(n-k, k) * 2^(n-k).
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2) * 2^((n+k)/2) * (1+(-1)^(n-k))/2. (End)
For asymptotics, see the Robinson paper. [This is disputed by Yen-chi R. Lin. See below, Sep 30 2013.]
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * C(n,2*k) * (2*k)!/k!. - Paul Barry, Feb 11 2008
G.f.: 1/(1 - 2*x - 2*x^2/(1 - 2*x - 4*x^2/(1 - 2*x - 6*x^2/(1 - 2*x - 8*x^2/(1 - ... (continued fraction). - Paul Barry, Feb 25 2010
E.g.f.: exp(x^2 + 2*x) = Q(0); Q(k) = 1 + (x^2 + 2*x)/(2*k + 1 - (x^2 + 2*x)*(2*k + 1)/((x^2 + 2*x) + (2*k + 2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: 1/Q(0), where Q(k) = 1 + 2*x*k - x - x/(1 - 2*x*(k + 1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
a(n) = (2*n/e)^(n/2) * exp(sqrt(2*n)) / sqrt(2*e) * (1 + sqrt(2/n)/3 + O(n^(-1))). - Yen-chi R. Lin, Sep 30 2013
0 = a(n)*(2*a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-2*a(n+1) + a(n+2)) for all n >= 0. - Michael Somos, Oct 23 2015
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k)*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

More terms from Larry Reeves (larryr(AT)acm.org), Feb 21 2001
Initial condition a(0)=1 added to definition by Jon E. Schoenfield, Oct 01 2013
More terms from Joerg Arndt, Oct 04 2013

A006330 Number of corners, or planar partitions of n with only one row and one column.

Original entry on oeis.org

1, 1, 3, 6, 12, 21, 38, 63, 106, 170, 272, 422, 653, 986, 1482, 2191, 3218, 4666, 6726, 9592, 13602, 19122, 26733, 37102, 51232, 70292, 95989, 130356, 176246, 237120, 317724, 423840, 563266, 745562, 983384, 1292333, 1692790, 2209886, 2876132

Offset: 0

N. J. A. Sloane

nonn

The first four terms a(0), a(1), a(2), a(3) agree with sequence A000219 for unrestricted planar partitions, since the restriction does not rule anything out. For a(4) there is just one planar partition which doesn't satisfy the restriction, four 1's arranged in a square. So A000219 has fifth term 13 and here we have 12.
a(n) + A001523(n) = A000712(n). - Michael Somos, Jul 22 2003
Number of unimodal compositions of n+1 where the maximal part appears once, see example. [Joerg Arndt, Jun 11 2013]

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(4)=12 unimodal compositions of 4+1=5 where the maximal part appears once:
01:  [ 1 1 1 2 ]
02:  [ 1 1 2 1 ]
03:  [ 1 1 3 ]
04:  [ 1 2 1 1 ]
05:  [ 1 3 1 ]
06:  [ 1 4 ]
07:  [ 2 1 1 1 ]
08:  [ 2 3 ]
09:  [ 3 1 1 ]
10:  [ 3 2 ]
11:  [ 4 1 ]
12:  [ 5 ]
(End)
G.f. = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 21*x^5 + 38*x^6 + 63*x^7 + 106*x^8 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see page 77.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See (5.6).
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)
G. Kreweras, Letter to N. J. A. Sloane

Cf. A000219, A000712, A197870.
Column k=1 of A247255.
Row sums of A259100.

a[0] = 1; a[n_] := SeriesCoefficient[ Sum[x^k/Product[1 - x^i, {i, 1, k}]^2, {k, 1, n}] + 1, {x, 0, n}]; Array[a, 39, 0] (* Jean-François Alcover, Mar 13 2014 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / prod(i=1, k, 1 - x^i, 1 + x*O(x^n))^2, 1), n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) - 1)\2, (-1)^k * x^((k + k^2)/2)) / eta(x + x*O(x^n))^2, n))};

G.f.: 1+Sum_{k>0} x^k/(Product_{i=1..k} (1-x^i))^2.
G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1 - x^k)^2. - Michael Somos, Jul 28 2003
Convolution product of A197870 and A000712. - Michael Somos, Feb 22 2015
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015

Edited and extended by Moshe Shmuel Newman, Jun 10 2003

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 7, 0, 1, 6, 20, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0

Offset: 0

Alois P. Heinz, Sep 09 2008

A(n,k) is also the number of partitions of n into parts of k kinds.
In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - Geoffrey Critzer, Nov 11 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5, ...
  0,   2,   5,   9,  14,  20, ...
  0,   3,  10,  22,  40,  65, ...
  0,   5,  20,  51, 105, 190, ...
  0,   7,  36, 108, 252, 506, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1.
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. 8.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.
Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).
Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.
Main diagonal gives A008485.
Antidiagonal sums give A067687.
Cf. A060642, A122768, A246935, A255961, A261718.

# DedekindEta is defined in A000594.
A144064Column(k, len) = DedekindEta(len, -k)
for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[0, ] = 1; a[, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]} ]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ M. F. Hasler, Aug 04 2024

G.f. of column k: Product_{j>=1} 1/(1-x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):

cp's OEIS Frontend

A002720 Number of partial permutations of an n-set; number of n X n binary matrices with at most one 1 in each row and column.

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A103919 Triangle of numbers of partitions of n with total number of odd parts equal to k from {0,...,n}.

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A001523 Number of stacks, or planar partitions of n; also weakly unimodal compositions of n.

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A115994 Triangle read by rows: T(n,k) is number of partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

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A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

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A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.