A080599 -id:A080599 - cp's OEIS frontend

A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, 211799312, 997313824, 4809701440, 23758664096, 119952692896, 618884638912, 3257843882624, 17492190577600, 95680443760576, 532985208200576, 3020676745975552

Offset: 0

N. J. A. Sloane and J. H. Conway

a(n) is also the number of n X n symmetric permutation matrices.
a(n) is also the number of matchings (Hosoya index) in the complete graph K(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
a(n) is also the number of independent vertex sets and vertex covers in the n-triangular graph. - Eric W. Weisstein, May 22 2017
Equivalently, this is the number of graphs on n labeled nodes with degrees at most 1. - Don Knuth, Mar 31 2008
a(n) is also the sum of the degrees of the irreducible representations of the symmetric group S_n. - Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
a(n) is the number of partitions of a set of n distinguishable elements into sets of size 1 and 2. - Karol A. Penson, Apr 22 2003
Number of tableaux on the edges of the star graph of order n, S_n (sometimes T_n). - Roberto E. Martinez II, Jan 09 2002
The Hankel transform of this sequence is A000178 (superfactorials). Sequence is also binomial transform of the sequence 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, ... (A001147 with interpolated zeros). - Philippe Deléham, Jun 10 2005
Row sums of the exponential Riordan array (e^(x^2/2),x). - Paul Barry, Jan 12 2006
a(n) is the number of nonnegative lattice paths of upsteps U = (1,1) and downsteps D = (1,-1) that start at the origin and end on the vertical line x = n in which each downstep (if any) is marked with an integer between 1 and the height of its initial vertex above the x-axis. For example, with the required integer immediately preceding each downstep, a(3) = 4 counts UUU, UU1D, UU2D, U1DU. - David Callan, Mar 07 2006
Equals row sums of triangle A152736 starting with offset 1. - Gary W. Adamson, Dec 12 2008
Proof of the recurrence relation a(n) = a(n-1) + (n-1)*a(n-2): number of involutions of [n] containing n as a fixed point is a(n-1); number of involutions of [n] containing n in some cycle (j, n), where 1 <= j <= n-1, is (n-1) times the number of involutions of [n] containing the cycle (n-1 n) = (n-1)*a(n-2). - Emeric Deutsch, Jun 08 2009
Number of ballot sequences (or lattice permutations) of length n. A ballot sequence B is a string such that, for all prefixes P of B, h(i) >= h(j) for i < j, where h(x) is the number of times x appears in P. For example, the ballot sequences of length 4 are 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1231, and 1234. The string 1221 does not appear in the list because in the 3-prefix 122 there are two 2's but only one 1. (Cf. p. 176 of Bruce E. Sagan: "The Symmetric Group"). - Joerg Arndt, Jun 28 2009
Number of standard Young tableaux of size n; the ballot sequences are obtained as a length-n vector v where v_k is the (number of the) row in which the number r occurs in the tableaux. - Joerg Arndt, Jul 29 2012
Number of factorial numbers of length n-1 with no adjacent nonzero digits. For example the 10 such numbers (in rising factorial radix) of length 3 are 000, 001, 002, 003, 010, 020, 100, 101, 102, and 103. - Joerg Arndt, Nov 11 2012
Also called restricted Stirling numbers of the second kind (see Mezo). - N. J. A. Sloane, Nov 27 2013
a(n) is the number of permutations of [n] that avoid the consecutive patterns 123 and 132. Proof. Write a self-inverse permutation in standard cycle form: smallest entry in each cycle in first position, first entries decreasing. For example, (6,7)(3,4)(2)(1,5) is in standard cycle form. Then erase parentheses. This is a bijection to the permutations that avoid consecutive 123 and 132 patterns. - David Callan, Aug 27 2014
Getu (1991) says these numbers are also known as "telephone numbers". - N. J. A. Sloane, Nov 23 2015
a(n) is the number of elements x in the symmetric group S_n such that x^2 = e where e is the identity. - Jianing Song, Aug 22 2018 [Edited on Jul 24 2025]
a(n) is the number of congruence orbits of upper-triangular n X n matrices on skew-symmetric matrices, or the number of Borel orbits in largest sect of the type DIII symmetric space SO_{2n}(C)/GL_n(C). Involutions can also be thought of as fixed-point-free partial involutions. See [Bingham and Ugurlu] link. - Aram Bingham, Feb 08 2020
From Thomas Anton, Apr 20 2020: (Start)
Apparently a(n) = b*c where b is odd iff a(n+b) (when a(n) is defined) is divisible by b.
Apparently a(n) = 2^(f(n mod 4)+floor(n/4))*q where f:{0,1,2,3}->{0,1,2} is given by f(0),f(1)=0, f(2)=1 and f(3)=2 and q is odd. (End)
From Iosif Pinelis, Mar 12 2021: (Start)
a(n) is the n-th initial moment of the normal distribution with mean 1 and variance 1. This follows because the moment generating function of that distribution is the e.g.f. of the sequence of the a(n)'s.
The recurrence a(n) = a(n-1) + (n-1)*a(n-2) also follows, by writing E(Z+1)^n=EZ(Z+1)^(n-1)+E(Z+1)^(n-1), where Z is a standard normal random variable, and then taking the first of the latter two integrals by parts. (End)

			Sequence starts 1, 1, 2, 4, 10, ... because possibilities are {}, {A}, {AB, BA}, {ABC, ACB, BAC, CBA}, {ABCD, ABDC, ACBD, ADCB, BACD, BADC, CBAD, CDAB, DBCA, DCBA}. - _Henry Bottomley_, Jan 16 2001
G.f. = 1 + x + 2*x^2 + 4*x^4 + 10*x^5 + 26*x^6 + 76*x^7 + 232*x^8 + 764*x^9 + ...
From _Gus Wiseman_, Jan 08 2021: (Start)
The a(4) = 10 standard Young tableaux:
  1 2 3 4
.
  1 2   1 3   1 2 3   1 2 4   1 3 4
  3 4   2 4   4       3       2
.
  1 2   1 3   1 4
  3     2     2
  4     4     3
.
  1
  2
  3
  4
The a(0) = 1 through a(4) = 10 set partitions into singletons or pairs:
  {}  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{3,4}}
             {{1},{2}}  {{1,2},{3}}    {{1,3},{2,4}}
                        {{1,3},{2}}    {{1,4},{2,3}}
                        {{1},{2},{3}}  {{1},{2},{3,4}}
                                       {{1},{2,3},{4}}
                                       {{1,2},{3},{4}}
                                       {{1},{2,4},{3}}
                                       {{1,3},{2},{4}}
                                       {{1,4},{2},{3}}
                                       {{1},{2},{3},{4}}
(End)

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Eric Weisstein's World of Mathematics, Hosoya Index
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Wikipedia, Telephone numbers.
Wikipedia, Young tableau.
Index entries for "core" sequences
Index entries for sequences related to Young tableaux.
Index entries for related partition-counting sequences

Cf. A001147, A001470, A053529, A099174, A136281-A136283, A001680, A001681.
See also A005425 for another version of the switchboard problem.
Equals 2 * A001475(n-1) for n>1.
First column of array A099020.
A069943(n+1)/A069944(n+1) = a(n)/A000142(n) in lowest terms.
Cf. A152736, A128229. - Gary W. Adamson, Dec 12 2008
Diagonal of A182172. - Alois P. Heinz, May 30 2012
Cf. A001813, A006882, A135401, A297708. - Manfred Scheucher, Jan 07 2018
Row sums of: A047884, A049403, A096713 (absolute value), A100861, A104556 (absolute value), A111924, A117506 (M_4 numbers), A122848, A238123.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A000700, A000701, A321729, A321730, A321737.

a000085 n = a000085_list !! n
  a000085_list = 1 : 1 : zipWith (+)
    (zipWith (*) [1..] a000085_list) (tail a000085_list) -- Reinhard Zumkeller, May 16 2013

A000085 := proc(n) option remember; if n=0 then 1 elif n=1 then 1 else procname(n-1)+(n-1)*procname(n-2); fi; end;
with(combstruct):ZL3:=[S,{S=Set(Cycle(Z,card<3))}, labeled]:seq(count(ZL3,size=n),n=0..25); # Zerinvary Lajos, Sep 24 2007
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, m>=card))}, labeled]; end: A:=a(2):seq(count(A, size=n), n=0..25); # Zerinvary Lajos, Jun 11 2008

<Roger L. Bagula, Oct 06 2006 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 28 2013 *)
a[ n_] := Sum[(2 k - 1)!! Binomial[ n, 2 k], {k, 0, n/2}]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, HypergeometricU[ -n/2, 1/2, -1/2] / (-1/2)^(n/2)]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ x + x^2 / 2], {x, 0, n}]]; (* Michael Somos, Jun 01 2013 *)
Table[(I/Sqrt[2])^n HermiteH[n, -I/Sqrt[2]], {n, 0, 100}] (* Emanuele Munarini, Mar 02 2016 *)
a[n_] := Sum[StirlingS1[n, k]*2^k*BellB[k, 1/2], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 18 2017, after Emanuele Munarini *)
RecurrenceTable[{a[n] == a[n-1] + (n-1)*a[n-2], a[0] == 1, a[1] == 1}, a, {n, 0, 20}] (* Joan Ludevid, Jun 17 2022 *)
sds[{}]:={{}};sds[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sds[Complement[set,s]]]/@Cases[Subsets[set,{1,2}],{i,_}]; Table[Length[sds[Range[n]]],{n,0,10}] (* Gus Wiseman, Jan 11 2021 *)

B(n,x):=sum(stirling2(n,k)*x^k,k,0,n);
  a(n):=sum(stirling1(n,k)*2^k*B(k,1/2),k,0,n);
  makelist(a(n),n,0,40); /* Emanuele Munarini, May 16 2014 */

makelist((%i/sqrt(2))^n*hermite(n,-%i/sqrt(2)),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp( x + x^2 / 2 + x * O(x^n)), n))}; /* Michael Somos, Nov 15 2002 */

N=66; x='x+O('x^N); egf=exp(x+x^2/2); Vec(serlaplace(egf)) \\ Joerg Arndt, Mar 07 2013

from math import factorial
def A000085(n): return sum(factorial(n)//(factorial(n-(k<<1))*factorial(k)*(1<>1)+1)) # Chai Wah Wu, Aug 31 2023

A000085 = lambda n: hypergeometric([-n/2,(1-n)/2], [], 2)
[simplify(A000085(n)) for n in range(28)] # Peter Luschny, Aug 21 2014

def a85(n): return sum(factorial(n) / (factorial(n-2*k) * 2**k * factorial(k)) for k in range(1+n//2))
for n in range(100): print(n, a85(n)) # Manfred Scheucher, Jan 07 2018

D-finite with recurrence a(0) = a(1) = 1, a(n) = a(n-1) + (n-1)*a(n-2) for n>1.
E.g.f.: exp(x+x^2/2).
a(n) = a(n-1) + A013989(n-2) = A013989(n)/(n+1) = 1+A001189(n).
a(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)!*2^k*k!).
a(m+n) = Sum_{k>=0} k!*binomial(m, k)*binomial(n, k)*a(m-k)*a(n-k). - Philippe Deléham, Mar 05 2004
For n>1, a(n) = 2*(A000900(n) + A000902(floor(n/2))). - Max Alekseyev, Oct 31 2015
The e.g.f. y(x) satisfies y^2 = y''y' - (y')^2.
a(n) ~ c*(n/e)^(n/2)exp(n^(1/2)) where c=2^(-1/2)exp(-1/4). [Chowla]
a(n) = HermiteH(n, 1/(sqrt(2)*i))/(-sqrt(2)*i)^n, where HermiteH are the Hermite polynomials. - Karol A. Penson, May 16 2002
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - Paul Barry, Jan 12 2006
For asymptotics see the Robinson paper.
a(n) = Sum_{m=0..n} A099174(n,m). - Roger L. Bagula, Oct 06 2006
O.g.f.: A(x) = 1/(1-x-1*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
From Gary W. Adamson, Dec 29 2008: (Start)
a(n) = (n-1)*a(n-2) + a(n-1); e.g., a(7) = 232 = 6*26 + 76.
Starting with offset 1 = eigensequence of triangle A128229. (End)
a(n) = (1/sqrt(2*Pi))*Integral_{x=-oo..oo} exp(-x^2/2)*(x+1)^n. - Groux Roland, Mar 14 2011
Row sums of |A096713|. a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A047974 and A080599. - Peter Bala, Dec 07 2011
From Sergei N. Gladkovskii, Dec 03 2011 - Oct 28 2013: (Start)
Continued fractions:
E.g.f.: 1+x*(2+x)/(2*G(0)-x*(2+x)) where G(k)=1+x*(x+2)/(2+2*(k+1)/G(k+1)).
G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1) - x*(k+1)/(1 - x/U(k+1)).
G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - x*(k+1)/Q(k+1)).
G.f.: -1/(x*Q(0)) where Q(k) = 1 - 1/x - (k+1)/Q(k+1).
G.f.: T(0)/(1-x) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x)^2/T(k+1)). (End)
a(n) ~ (1/sqrt(2)) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014
a(n) = Sum_{k=0..n} s(n,k)*(-1)^(n-k)*2^k*B(k,1/2), where the s(n,k) are (signless) Stirling numbers of the first kind, and the B(n,x) = Sum_{k=0..n} S(n,k)*x^k are the Stirling polynomials, where the S(n,k) are the Stirling numbers of the second kind. - Emanuele Munarini, May 16 2014
a(n) = hyper2F0([-n/2,(1-n)/2],[],2). - Peter Luschny, Aug 21 2014
0 = a(n)*(+a(n+1) + a(n+2) - a(n+3)) + a(n+1)*(-a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Aug 22 2014
From Peter Bala, Oct 06 2021: (Start)
a(n+k) == a(n) (mod k) for all n >= 0 and all positive odd integers k.
Hence for each odd k, the sequence obtained by taking a(n) modulo k is a periodic sequence and the exact period divides k. For example, taking a(n) modulo 7 gives the purely periodic sequence [1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, 1, 1, 2, 4, 3, 5, 6, ...] of period 7. For similar results see A047974 and A115329. (End)

A005442 a(n) = n!*Fibonacci(n+1).

Original entry on oeis.org

1, 1, 4, 18, 120, 960, 9360, 105840, 1370880, 19958400, 322963200, 5748019200, 111607372800, 2347586841600, 53178757632000, 1290674601216000, 33413695451136000, 919096314200064000, 26768324463648768000

Offset: 0

Simon Plouffe

Number of ways to use the elements of {1,...,n} once each to form a sequence of lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005
Number of pairs (p,S) where p is a permutation of {1,2,...,n} and S is a subset of {1,2,...,n} such that if s is in S then p(s) is not in S.  For example a(2) = 4 because we have (p=(1)(2), s={}); (p=(1,2), s={}); (p=(1,2), s={1}); (p=(1,2), s={2}) where p is given in cycle notation. - Geoffrey Critzer, Jul 01 2013
Another way to state the first comment: a(n) is the number of ways to partition [n] into blocks of size at most 2, order the blocks, and order the elements within each block. For example, a(5)=960 since there are 3 cases: 1) 1,2,3,4,5: 120 such ordered blocks, 1 way to order elements within blocks, hence 120 ways; 2) 12,3,4,5: 240 such ordered blocks, 2 ways to order elements within blocks, hence 480 ways; 3) 12,34,5: 90 such ordered blocks, 4 ways to order elements within blocks, hence 360 ways. - Enrique Navarrete, Sep 01 2023

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

Seiichi Manyama, Table of n, a(n) for n = 0..416
P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, A family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 81-83.
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.
P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 494
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Index entries for related partition-counting sequences

Row sums of Fibonacci Jabotinsky-triangle A039692.

Cf. A000045, A000142, A039948, A052585, A080599.

[Factorial(n)*Fibonacci(n+1): n in [0..20]]; // G. C. Greubel, Nov 20 2022

Table[Fibonacci[n + 1]*n!, {n, 0, 20}] (* Zerinvary Lajos, Jul 09 2009 *)

a(n) = n!*fibonacci(n+1) \\ Charles R Greathouse IV, Oct 03 2016

[fibonacci(n+1)*factorial(n) for n in range(21)] # G. C. Greubel, Nov 20 2022

a(n) = A039948(n,0).
E.g.f.: 1/(1-x-x^2).
D-finite with recurrence a(n) = n*a(n-1)+n*(n-1)*a(n-2). - Detlef Pauly (dettodet(AT)yahoo.de), Sep 22 2003
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. Cf. A080599 and A052585. - Peter Bala, Dec 07 2011

Comments from Wolfdieter Lang

A122704 a(n) = Sum_{k=0..n} 3^(n-k)*A123125(n, k).

Original entry on oeis.org

1, 1, 4, 22, 160, 1456, 15904, 202672, 2951680, 48361216, 880405504, 17630351872, 385148108800, 9114999832576, 232311251144704, 6343764407375872, 184778982658539520, 5718564661248065536, 187389427488113557504, 6481629887083387420672, 235993351028007334051840

Offset: 0

Philippe Deléham, Oct 22 2006

a(n+1) = [1,4,22,160,1456,...] is the first Eulerian transform of A000244 (powers of 3), it is also the Stirling transform of A080599(n+1) = [1,3,12,66,450,...].

			G.f. = 1 + x + 4*x^2 + 22*x^3 + 160*x^4 + 1456*x^5 + 15904*x^6 + ... - _Michael Somos_, Jun 05 2021

Alois P. Heinz, Table of n, a(n) for n = 0..409 (first 101 terms from G. C. Greubel)
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
T. J. Stieltjes, Sur quelques intégrales définies et leur développement en fractions continues, Q. J. Math., London, 24, 1890, pp. 370-382.
T. J. Stieltjes, Sur quelques intégrales définies et leur développement en fractions continues, LXXVII, p.382, Stieltjes T.J. Oeuvres complètes, tome 2, Noordhoff, 1918, 617p.
OEIS Wiki, Eulerian polynomials.
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A076726.

# From Peter Luschny, Jun 27 2019: (Start)
seq(subs(x=3, add(combinat:-eulerian1(n,k)*x^k, k=0..n)), n=0..20);
# Alternative:
gf := -2/(exp(2*x) - 3): series(gf, x, 21): seq(n!*coeff(%, x, n), n=0..20);
# (End)
# Or via the recurrence of the Fubini polynomials:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n - k)*x, k = 1..n)) end:
a := n -> 2^n*subs(x = 1/2, F(n)):
seq(a(n), n = 0..20); # Peter Luschny, May 21 2021
# next Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*2^(j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 30 2021

CoefficientList[Series[(Exp[x]-2*Cosh[x])/(2*Exp[x]-3*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 24 2013 *)
Table[Sum[2^(n+1)*k^n/3^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 28 2013 *)
Round@Table[(-1)^(n+1) (PolyLog[-n, Sqrt[3]] + PolyLog[-n, -Sqrt[3]])/3, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)
Table[Sum[StirlingS2[n, k]*2^(n-k)*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 13 2018 *)
Eulerian1[0, 0] = 1; Eulerian1[n_, k_] := Sum[(-1)^j (k-j+1)^n Binomial[ n+1, j], {j, 0, k+1}]; Table[Sum[Eulerian1[n, k]*3^k, {k, 0, n}], {n, 0, 20}] (* Jean-François Alcover, Jul 13 2019, after Peter Luschny *)
a[n_] := (-2)^(n + 1) PolyLog[-n, 3] / 3;
Table[a[n], {n, 0, 20}] (* Peter Luschny, Aug 20 2021 *)

{a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1-2*k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

{a(n) = if(n<0, 0, n!*polcoeff( 2/(3 - exp(2*x + x*O(x^n))), n))}; /* Michael Somos, Jun 05 2021 */

O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1-2*k*x). - Paul D. Hanna, Jul 20 2011
a(n) = Sum_{k=0..n} A131689(n,k)*2^(n-k). - Philippe Deléham, Oct 09 2007
a(n) = A_{n}(3) where A_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010
E.g.f.: (exp(x) - 2*cosh(x))/(2*exp(x) - 3*cosh(x)) =1 + x/(U(0)-x) where U(k)=  4*k+1 - x/(1 + x/(4*k+3 - x/(1 + x/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 08 2012
G.f.: 1 + x/G(0) where G(k) = 1 - x*2*(2*k+2) + x^2*(k+1)*(k+2)*(1-2^2)/G(k+1); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Jan 11 2013
a(n) ~ n!/3 * (2/log(3))^(n+1). - Vaclav Kotesovec, Jun 24 2013
G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+1) - 3*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = Sum_{k>=0} 2^(n+1)*k^n/3^(k+1). - Vaclav Kotesovec, Nov 28 2013
a(n) = 2^n*log(3)* Integral_{x >= 0} (floor(x))^n * 3^(-x) dx. - Peter Bala, Feb 14 2015
From Karol A. Penson, Sep 04 2015: (Start)
E.g.f.: 2/(3-exp(2*x)).
Special values of the generalized hypergeometric function n_F_(n-1):
a(n) = (2^(n+1)/9) * hypergeom([2,2,..2],[1,1,..1],1/3), where the sequence in the first square bracket ("upper" parameters) has n elements all equal to 2 whereas the sequence in the second square bracket ("lower" parameters) has n-1 elements all equal to 1.
Example: a(4) = (2^5/9) * hypergeom([2,2,2,2],[1,1,1],1/3) = 16. (End)
a(n) = (-1)^(n+1)*(Li_{-n}(sqrt(3)) + Li_{-n}(-sqrt(3)))/3, where Li_n(x) is the polylogarithm. - Vladimir Reshetnikov, Oct 31 2015
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 2^(k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 16 2020
a(n) = 2^n*F_{n}(1/2), where F_{n}(x) is the Fubini polynomial. This is another way to state the above formula from Ilya Gutkovskiy. - Peter Luschny, May 21 2021
a(n+1) = -2*a(n) + 3*Sum_{k=0..n} binomial(n, k)*a(k)*a(n-k). - Michael Somos, Jun 05 2021
a(n) = (-2)^(n + 1)*PolyLog(-n, 3)/3. - Peter Luschny, Aug 20 2021

a(7) corrected (was 206672) and more terms from Peter Luschny, Aug 03 2010

More terms from Vaclav Kotesovec, Jul 13 2018

A339888 Number of non-isomorphic multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 3, 5, 13, 23, 55, 104, 236, 470, 1039, 2140, 4712, 9962, 21961, 47484, 105464, 232324, 521338, 1167825, 2651453, 6031136, 13863054, 31987058, 74448415, 174109134, 410265423, 971839195, 2317827540, 5558092098, 13412360692, 32542049038, 79424450486

Offset: 0

Gus Wiseman, Jan 09 2021

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 13 multiset partitions:
  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{1,2}}
         {{1},{1}}  {{2},{1,2}}    {{1,2},{3,4}}
         {{1},{2}}  {{1},{1},{1}}  {{1,3},{2,3}}
                    {{1},{2},{2}}  {{1},{1},{2,3}}
                    {{1},{2},{3}}  {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The version for set partitions is A000085, with ordered version A080599.
The case of integer partitions is 1 + A004526(n), ranked by A003586.
Non-isomorphic multiset partitions are counted by A007716.
The case without singletons is A007717.
The version allowing non-strict pairs (x,x) is A320663.
A001190 counts rooted trees with out-degrees <= 2, ranked by A292050.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
A339887 counts factorizations into primes or squarefree semiprimes.
Cf. A001055, A007718, A316983, A319616, A320656, A321729, A339740, A339741.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
gs(v) = {sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i], v[j])); g*x^(2*v[i]*v[j]/g))) + sum(i=1, #v, my(r=v[i]); (1 + (1+r)%2)*x^r + ((r-1)\2)*x^(2*r))}
a(n)={if(n==0, 1, my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(gs(p) + O(x*x^n), -n))[n]); s/n!)} \\ Andrew Howroyd, Apr 16 2021

Terms a(11) and beyond from Andrew Howroyd, Apr 16 2021

A232432 Number of compositions of n avoiding the pattern 111.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 21, 34, 59, 114, 178, 284, 522, 823, 1352, 2133, 3739, 5807, 9063, 14074, 23639, 36006, 56914, 87296, 131142, 214933, 324644, 487659, 739291, 1108457, 1724673, 2558386, 3879335, 5772348, 8471344, 12413666, 19109304, 27886339, 40816496

Offset: 0

Alois P. Heinz, Nov 23 2013

nonn

Number of compositions of n into parts with multiplicity <= 2.

			a(4) = 7: [4], [3,1], [2,2], [1,3], [2,1,1], [1,2,1], [1,1,2].
a(5) = 11: [5], [4,1], [3,2], [2,3], [1,4], [3,1,1], [2,2,1], [1,3,1], [2,1,2], [1,2,2], [1,1,3].
a(6) = 21: [6], [4,2], [3,3], [5,1], [2,4], [1,5], [2,1,3], [1,2,3], [1,1,4], [4,1,1], [3,2,1], [2,3,1], [1,4,1], [3,1,2], [1,3,2], [1,2,2,1], [2,1,1,2], [1,2,1,2], [1,1,2,2], [2,2,1,1], [2,1,2,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A000726 (partitions avoiding 111), A032020 (pattern 11), A128695 (adjacent pattern 111).
Column k=2 of A243081.
The case of partitions is ranked by A004709.
The version for patterns is A080599.
(1,1,1,1)-avoiding partitions are counted by A232464.
The (1,1,1)-matching version is A335455.
Patterns matched by compositions are counted by A335456.
The version for prime indices is A335511.
(1,1,1)-avoiding compositions are ranked by A335513.
Cf. A011782, A006918, A102726, A106356, A238279, A333755.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j)/j!, j=0..min(n/i, 2))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);

f[list_]:=Apply[And,Table[Count[list,i]<3,{i,1,Max[list]}]];
g[list_]:=Length[list]!/Apply[Times,Table[Count[list,i]!,{i,1,Max[list]}]];
a[n_] := If[n == 0, 1, Total[Map[g, Select[IntegerPartitions[n], f]]]];
Table[a[n], {n, 0, 35}] (* Geoffrey Critzer, Nov 25 2013, updated by Jean-François Alcover, Nov 20 2023 *)

A276921 Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 6, 0, 1, 1, 3, 12, 24, 0, 1, 1, 3, 13, 66, 120, 0, 1, 1, 3, 13, 74, 450, 720, 0, 1, 1, 3, 13, 75, 530, 3690, 5040, 0, 1, 1, 3, 13, 75, 540, 4550, 35280, 40320, 0, 1, 1, 3, 13, 75, 541, 4670, 45570, 385560, 362880, 0

Offset: 0

Alois P. Heinz, Sep 22 2016

			Square array A(n,k) begins:
  1,    1,     1,     1,     1,     1,     1,     1, ...
  0,    1,     1,     1,     1,     1,     1,     1, ...
  0,    2,     3,     3,     3,     3,     3,     3, ...
  0,    6,    12,    13,    13,    13,    13,    13, ...
  0,   24,    66,    74,    75,    75,    75,    75, ...
  0,  120,   450,   530,   540,   541,   541,   541, ...
  0,  720,  3690,  4550,  4670,  4682,  4683,  4683, ...
  0, 5040, 35280, 45570, 47110, 47278, 47292, 47293, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.

Columns k=0..10 give: A000007, A000142, A080599, A189886, A276924, A276925, A276926, A276927, A276928, A276929, A276930.
Main diagonal gives A000670.
Cf. A276922.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
       A(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

E.g.f. of column k: 1/(1-Sum_{i=1..k} x^i/i!).

A(n,k) = Sum_{j=0..k} A276922(n,j).

A335455 Number of compositions of n with some part appearing more than twice.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 11, 30, 69, 142, 334, 740, 1526, 3273, 6840, 14251, 29029, 59729, 122009, 248070, 500649, 1012570, 2040238, 4107008, 8257466, 16562283, 33229788, 66621205, 133478437, 267326999, 535146239, 1071183438, 2143604313, 4289194948, 8581463248

Offset: 0

Gus Wiseman, Jun 15 2020

nonn

Also the number of compositions of n matching the pattern (1,1,1).

A composition of n is a finite sequence of positive integers summing to n.

			The a(3) = 1 through a(6) = 11 compositions:
  (111)  (1111)  (1112)   (222)
                 (1121)   (1113)
                 (1211)   (1131)
                 (2111)   (1311)
                 (11111)  (3111)
                          (11112)
                          (11121)
                          (11211)
                          (12111)
                          (21111)
                          (111111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The case of partitions is A000726.
The avoiding version is A232432.
The (1,1)-matching version is A261982.
The version for patterns is A335508.
The version for prime indices is A335510.
These compositions are ranked by A335512.
Compositions are counted by A011782.
Combinatory separations are counted by A269134.
Normal patterns matched by compositions are counted by A335456.
Cf. A000740, A034691, A056986, A080599, A106356, A181796, A232464, A238279, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Max@@Length/@Split[Sort[#]]>=3&]],{n,0,10}]

a(n > 0) = 2^(n - 1) - A232432(n).

A052585 E.g.f. 1/(1-x-2*x^2).

Original entry on oeis.org

1, 1, 6, 30, 264, 2520, 30960, 428400, 6894720, 123742080, 2478470400, 54486432000, 1308153369600, 34005760588800, 952248474777600, 28566146568960000, 914137612996608000, 31080323154456576000, 1118898035934142464000, 42518003720397004800000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Laguerre transform is A052563. - Paul Barry, Aug 08 2008

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 530

Cf. A080599, A005442.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1 -x -2*x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

spec := [S,{S=Sequence(Union(Z,Prod(Z,Union(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{m = 50}, CoefficientList[Series[-1/(-1 + x + 2*x^2), {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 17 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1 -x -2*x^2))) \\ G. C. Greubel, May 17 2018

E.g.f.: 1/(1 -x -2*x^2).
Recurrence: a(1)=1, a(0)=1, (-2*n^2-6*n-4)*a(n)+(-2-n)*a(n+1)+a(n+2)=0.
a(n) = Sum(1/9*(1+4*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+2*_Z^2))*n!.
a(n) = n!*A001045(n+1). - Paul Barry, Aug 08 2008
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+8*x)*d/dx. Cf. A080599 and A005442. - Peter Bala, Dec 07 2011

a(18)-a(19) added by G. C. Greubel, May 17 2018

A335511 Number of (1,1,1)-avoiding permutations of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 0, 1, 2, 0, 3, 1, 6, 1, 0, 2, 2, 2, 6, 1, 2, 2, 0, 1, 6, 1, 3, 3, 2, 1, 0, 1, 3, 2, 3, 1, 0, 2, 0, 2, 2, 1, 12, 1, 2, 3, 0, 2, 6, 1, 3, 2, 6, 1, 0, 1, 2, 3, 3, 2, 6, 1, 0, 0, 2, 1, 12, 2, 2

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Patterns avoiding this pattern are counted by A080599.
These compositions are counted by A232432.
The (1,1)-avoiding version is A335451.
The complement A335510 is the matching version.
These permutations are ranked by A335513.
Patterns are counted by A000670 and ranked by A333217.
Permutations of prime indices are counted by A008480.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,_,x_,_,x_,_}]&]],{n,100}]

If n is cubefree, a(n) = A008480(n), otherwise a(n) = 0.

A335508 Number of patterns of length n matching the pattern (1,1,1).

Original entry on oeis.org

0, 0, 0, 1, 9, 91, 993, 12013, 160275, 2347141, 37496163, 649660573, 12142311195, 243626199181, 5224710549243, 119294328993853, 2889836999693355, 74037381200415901, 2000383612949821323, 56850708386783835133, 1695491518035158123115, 52949018580275965241821

Offset: 0

Gus Wiseman, Jun 18 2020

nonn

			The a(3) = 1 through a(4) = 9 patterns:
  (1,1,1)  (1,1,1,1)
           (1,1,1,2)
           (1,1,2,1)
           (1,2,1,1)
           (1,2,2,2)
           (2,1,1,1)
           (2,1,2,2)
           (2,2,1,2)
           (2,2,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..424
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The complement A080599 is the avoiding version.
Permutations of prime indices matching this pattern are counted by A335510.
Compositions matching this pattern are counted by A335455 and ranked by A335512.
Patterns are counted by A000670 and ranked by A333217.
Patterns matching the pattern (1,1) are counted by A019472.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Patterns matching (1,2,3) are counted by A335515.
Cf. A034691, A056986, A232464, A238279, A292884, A333175, A333755, A335451, A335456, A335457, A335458.
Cf. A276922.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
a:= n-> b(n$2)-b(n, 2):
seq(a(n), n=0..21);  # Alois P. Heinz, Jan 28 2024

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],MatchQ[#,{_,x_,_,x_,_,x_,_}]&]],{n,0,6}]

a(n) = Sum_{k=3..n} A276922(n,k). - Alois P. Heinz, Jan 28 2024

a(n) = A000670(n) - A080599(n). - Andrew Howroyd, Jan 28 2024

a(9)-a(21) from Alois P. Heinz, Jan 28 2024

cp's OEIS Frontend

A000085 Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

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A005442 a(n) = n!*Fibonacci(n+1).

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A122704 a(n) = Sum_{k=0..n} 3^(n-k)*A123125(n, k).

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A339888 Number of non-isomorphic multiset partitions of weight n into singletons or strict pairs.

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A232432 Number of compositions of n avoiding the pattern 111.

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A276921 Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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