A000670 -id:A000670 - cp's OEIS frontend

A019536 Number of length n necklaces with integer entries that cover an initial interval of positive integers.

1, 2, 5, 20, 109, 784, 6757, 68240, 787477, 10224812, 147512053, 2340964372, 40527565261, 760095929840, 15352212731933, 332228417657960, 7668868648772701, 188085259070219000, 4884294069438337429

Offset: 1

Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)

Original name: a(n) = number of necklaces of n beads with up to n unlabeled colors.

The Moebius transform of this sequence is A060223.

			a(3) = 5 since there are the following length 3 words up to rotation:
     111,  112, 122, 123, 132.
a(4) = 20 since there are the following length 4 words up to rotation:
     1111,
     1112, 1122, 1212, 1222,
     1123, 1132, 1213, 1223, 1232, 1233, 1322, 1323, 1332,
     1234, 1243, 1324, 1342, 1423, 1432.

G. C. Greubel, Table of n, a(n) for n = 1..420
M. Goebel, On the number of special permutation-invariant orbits and terms, in: Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.); see p. 509 (stated as an open problem).
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Eric Weisstein's world of Mathematics, Necklaces.

Cf. A000670, A008277, A019537, A060223.

Row sums of A087854. - Philippe Deléham

Needs["DiscreteMath`Combinatorica`"];
mult[li:{__Integer}] := Multinomial @@ Length /@ Split[Sort[li]];
neck[li:{__Integer}] := Module[{n, d}, n=Plus @@ li; d=n-First[li];Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times @@ ((li/#2)!)&, 0, Divisors[GCD @@ li]]/n];
Table[(mult /@ Partitions[n]).(neck /@ Partitions[n]), {n, 24}]
(* second program: *)
a[n_] := Sum[DivisorSum[n, EulerPhi[#]*StirlingS2[n/#, k] k! &]/n, {k, 1, n}];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 31 2016, after Philippe Deléham *)

a(n) = sum(k=1, n, sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2)*k!)/n); \\ Michel Marcus, Mar 31 2016

See Mathematica code.
a(n) ~ (n-1)! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Jul 21 2019
From Petros Hadjicostas, Aug 19 2019: (Start)
The first formula is due to Philippe Deléham from the Crossrefs (see also the programs below). The second one follows easily from the first one. The third one follows from the second one using the associative property of Dirichlet convolutions.
a(n) = Sum_{k = 1..n} (k!/n) * Sum_{d|n} phi(d) * S2(n/d, k), where S2(n, k) = Stirling numbers of 2nd kind (A008277).
a(n) = (1/n) * Sum_{d|n} phi(d) * A000670(n/d).
a(n) = Sum_{d|n} A060223(d).
(End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = (1/n)*Sum_{k=1..n} A000670(gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} A000670(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Edited by Wouter Meeussen, Aug 06 2002
Corrected by T. D. Noe, Oct 31 2006
Edited by Andrew Howroyd, Aug 19 2019

A094587 Triangle of permutation coefficients arranged with 1's on the diagonal. Also, triangle of permutations on n letters with exactly k+1 cycles and with the first k+1 letters in separate cycles.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 24, 24, 12, 4, 1, 120, 120, 60, 20, 5, 1, 720, 720, 360, 120, 30, 6, 1, 5040, 5040, 2520, 840, 210, 42, 7, 1, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 3628800, 3628800

Offset: 0

Paul Barry, May 13 2004

Also, table of Pochhammer sequences read by antidiagonals (see Rudolph-Lilith, 2015). - N. J. A. Sloane, Mar 31 2016
Reverse of A008279. Row sums are A000522. Diagonal sums are A003470. Rows of inverse matrix begin {1}, {-1,1}, {0,-2,1}, {0,0,-3,1}, {0,0,0,-4,1} ... The signed lower triangular matrix (-1)^(n+k)n!/k! has as row sums the signed rencontres numbers Sum_{k=0..n} (-1)^(n+k)n!/k!. (See A000166). It has matrix inverse 1 1,1 0,2,1 0,0,3,1 0,0,0,4,1,...
Exponential Riordan array [1/(1-x),x]; column k has e.g.f. x^k/(1-x). - Paul Barry, Mar 27 2007
From Tom Copeland, Nov 01 2007: (Start)
T is the umbral extension of n!*Lag[n,(.)!*Lag[.,x,-1],0] = (1-D)^(-1) x^n = (-1)^n * n! * Lag(n,x,-1-n) = Sum_{j=0..n} binomial(n,j) * j! * x^(n-j) = Sum_{j=0..n} (n!/j!) x^j. The inverse operator is A132013 with generalizations discussed in A132014.
b = T*a can be characterized several ways in terms of a(n) and b(n) or their o.g.f.'s A(x) and B(x).
1) b(n) = n! Lag[n,(.)!*Lag[.,a(.),-1],0], umbrally,
2) b(n) = (-1)^n n! Lag(n,a(.),-1-n)
3) b(n) = Sum_{j=0..n} (n!/j!) a(j)
4) B(x) = (1-xDx)^(-1) A(x), formally
5) B(x) = Sum_{j=0,1,...} (xDx)^j A(x)
6) B(x) = Sum_{j=0,1,...} x^j * D^j * x^j A(x)
7) B(x) = Sum_{j=0,1,...} j! * x^j * L(j,-:xD:,0) A(x) where Lag(n,x,m) are the Laguerre polynomials of order m, D the derivative w.r.t. x and (:xD:)^j = x^j * D^j. Truncating the operator series at the j = n term gives an o.g.f. for b(0) through b(n).
c = (0!,1!,2!,3!,4!,...) is the sequence associated to T under the list partition transform and the associated operations described in A133314 so T(n,k) = binomial(n,k)*c(n-k). The reciprocal sequence is d = (1,-1,0,0,0,...). (End)
From Peter Bala, Jul 10 2008: (Start)
This array is the particular case P(1,1) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below:
n\k|0.....................1...............2.......3......4
----------------------------------------------------------
0..|1.....................................................
1..|a....................1................................
2..|a(a+b)...............2a..............1................
3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
...
The entries A(n,k) of this array satisfy the recursion A(n,k) = (a+b*(n-k-1))*A(n-1,k) + A(n-1,k-1), which reduces to the Pascal formula when a = 1, b = 0.
Various cases are recorded in the database, including: P(1,0) = Pascal's triangle A007318, P(2,0) = A038207, P(3,0) = A027465, P(2,1) = A132159, P(1,3) = A136215 and P(2,3) = A136216.
When b <> 0 the array P(a,b) has e.g.f. exp(x*y)/(1-b*y)^(a/b) = 1 + (a+x)*y + (a*(a+b)+2a*x+x^2)*y^2/2! + (a*(a+b)*(a+2b) + 3a*(a+b)*x + 3a*x^2+x^3)*y^3/3! + ...; the array P(a,0) has e.g.f. exp((x+a)*y).
We have the matrix identities P(a,b)*P(a',b) = P(a+a',b); P(a,b)^-1 = P(-a,b).
An analog of the binomial expansion for the row entries of P(a,b) has been proved by [Echi]. Introduce a (generally noncommutative and nonassociative) product ** on the ring of polynomials in two variables by defining F(x,y)**G(x,y) = F(x,y)G(x,y) + by^2*d/dy(G(x,y)).
Define the iterated product F^(n)(x,y) of a polynomial F(x,y) by setting F^(1) = F(x,y) and F^(n)(x,y) = F(x,y)**F^(n-1)(x,y) for n >= 2. Then (x+a*y)^(n) = x^n + C(n,1)*a*x^(n-1)*y + C(n,2)*a*(a+b)*x^(n-2)*y^2 + ... + C(n,n)*a*(a+b)*(a+2b)*...*(a+(n-1)b)*y^n. (End)
(n+1) * n-th row = reversal of triangle A068424: (1; 2,2; 6,6,3; ...) - Gary W. Adamson, May 03 2009
Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n,k,p) = G(n-1,n-k,p) then T(n, k, 1) is this sequence, T(n, k, 2) = A112292(n, k) and T(n, k, 3) = A136214. - Peter Luschny, Jun 01 2009, revised Jun 18 2019
The higher order exponential integrals E(x,m,n) are defined in A163931. For a discussion of the asymptotic expansions of the E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + (n^2+n)/x^2 - (2*n+3*n^2+n^3)/x^3 + (6*n+11*n^2+6*n^3+n^4)/x^3 - ...) see A130534. The asymptotic expansion of E(x,m=1,n) leads for n >= 1 to the left hand columns of the triangle given above. Triangle A165674 is generated by the asymptotic expansions of E(x,m=2,n). - Johannes W. Meijer, Oct 07 2009
T(n,k) = n!/k! = number of permutations of [n+1] with exactly k+1 cycles and with elements 1,2,...,k+1 in separate cycles. See link and example below. - Dennis P. Walsh, Jan 24 2011
T(n,k) is the number of n permutations that leave some size k subset of {1,2,...,n} fixed. Sum_{k=0..n}(-1)^k*T(n,k) = A000166(n) (the derangements). - Geoffrey Critzer, Dec 11 2011
T(n,k) = A162995(n-1,k-1), 2 <= k <= n; T(n,k) = A173333(n,k), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012
The row polynomials form an Appell sequence. The matrix is a special case of a group of general matrices sketched in A132382. - Tom Copeland, Dec 03 2013
For interpretations in terms of colored necklaces, see A213936 and A173333. - Tom Copeland, Aug 18 2016
See A008279 for a relation of this entry to the e.g.f.s enumerating the faces of permutahedra and stellahedra. - Tom Copeland, Nov 14 2016
Also, T(n,k) is the number of ways to arrange n-k nonattacking rooks on the n X (n-k) chessboard. - Andrey Zabolotskiy, Dec 16 2016
The infinitesimal generator of this triangle is the generalized exponential Riordan array [-log(1-x), x] and equals the unsigned version of A238363. - Peter Bala, Feb 13 2017
Formulas for exponential and power series infinitesimal generators for this triangle T are given in Copeland's 2012 and 2014 formulas as T = unsigned exp[(I-A238385)] =  1/(I - A132440), where I is the identity matrix. - Tom Copeland, Jul 03 2017
If A(0) = 1/(1-x), and A(n) = d/dx(A(n-1)), then A(n) = n!/(1-x)^(n+1) = Sum_{k>=0} (n+k)!/k!*x^k = Sum_{k>=0} T(n+k, k)*x^k. - Michael Somos, Sep 19 2021

			Rows begin {1}, {1,1}, {2,2,1}, {6,6,3,1}, ...
For n=3 and k=1, T(3,1)=6 since there are exactly 6 permutations of {1,2,3,4} with exactly 2 cycles and with 1 and 2 in separate cycles. The permutations are (1)(2 3 4), (1)(2 4 3), (1 3)(2 4), (1 4)(2 3), (1 3 4)(2), and (1 4 3)(2). - _Dennis P. Walsh_, Jan 24 2011
Triangle begins:
     1,
     1,    1,
     2,    2,    1,
     6,    6,    3,    1,
    24,   24,   12,    4,    1,
   120,  120,   60,   20,    5,    1,
   720,  720,  360,  120,   30,    6,    1,
  5040, 5040, 2520,  840,  210,   42,    7,    1
The production matrix is:
      1,     1,
      1,     1,     1,
      2,     2,     1,    1,
      6,     6,     3,    1,    1,
     24,    24,    12,    4,    1,   1,
    120,   120,    60,   20,    5,   1,   1,
    720,   720,   360,  120,   30,   6,   1,   1,
   5040,  5040,  2520,  840,  210,  42,   7,   1,   1,
  40320, 40320, 20160, 6720, 1680, 336,  56,   8,   1,   1
which is the exponential Riordan array A094587, or [1/(1-x),x], with an extra superdiagonal of 1's.
Inverse begins:
   1,
  -1,  1,
   0, -2,  1,
   0,  0, -3,  1,
   0,  0,  0, -4,  1,
   0,  0,  0,  0, -5,  1,
   0,  0,  0,  0,  0, -6,  1,
   0,  0,  0,  0,  0,  0, -7,  1

Reinhard Zumkeller, Rows n = 0..149 of triangle, flattened
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 3.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 5.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 17.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) 11.6.7.
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator Part V, 2014.
T. Copeland, Compositional inverse operators and Sheffer sequences, 2016.
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
Othman Echi, Binomial coefficients and Nasir al-Din al-Tusi, Scientific Research and Essays Vol.1 (2), 28-32 November 2006.
H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010 (eqn. 10.35, p.49).
A. Hennessy and P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Peter Luschny, Variants of Variations.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
Dennis Walsh, A note on permutations with cyclic constraints
Wikipedia, Sheffer sequence

Cf. A000166 (alt. row sums), A000522 (row sums).
Cf. A068424, A036039, A173333, A213936, A263916.
Cf. A000670, A008279, A021009, A048994, A132013, A132014, A132159, A132440, A133314, A218234, A235706, A238385.

a094587 n k = a094587_tabl !! n !! k
a094587_row n = a094587_tabl !! n
a094587_tabl = map fst $ iterate f ([1], 1)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

T := proc(n, m): n!/m! end: seq(seq(T(n, m), m=0..n), n=0..9);  # Johannes W. Meijer, Oct 07 2009, revised Nov 25 2012
# Alternative: Note that if you leave out 'abs' you get A021009.
T := proc(n, k) option remember; if n = 0 and k = 0 then 1 elif k < 0 or k > n then 0 else abs((n + k)*T(n-1, k) - T(n-1, k-1)) fi end: #  Peter Luschny, Dec 30 2021

Flatten[Table[Table[n!/k!, {k,0,n}], {n,0,10}]] (* Geoffrey Critzer, Dec 11 2011 *)

def A094587_row(n): return (factorial(n)*exp(x).taylor(x,0,n)).list()
for n in (0..7): print(A094587_row(n)) # Peter Luschny, Sep 28 2017

T(n, k) = n!/k! if n >= k >= 0, otherwise 0.
T(n, k) = Sum_{i=k..n} |S1(n+1, i+1)*S2(i, k)| * (-1)^i, with S1, S2 the Stirling numbers.
T(n,k) = (n-k)*T(n-1,k) + T(n-1,k-1). E.g.f.: exp(x*y)/(1-y) = 1 + (1+x)*y + (2+2*x+x^2)*y^2/2! + (6+6*x+3*x^2+x^3)*y^3/3!+ ... . - Peter Bala, Jul 10 2008
A094587 = 1 / ((-1)*A129184 * A127648 + I), I = Identity matrix. - Gary W. Adamson, May 03 2009
From Johannes W. Meijer, Oct 07 2009: (Start)
The o.g.f. of right hand column k is Gf(z;k) = (k-1)!/(1-z)^k, k => 1.
The recurrence relations of the right hand columns lead to Pascal's triangle A007318. (End)
Let f(x) = (1/x)*exp(-x). The n-th row polynomial is R(n,x) = (-x)^n/f(x)*(d/dx)^n(f(x)), and satisfies the recurrence equation R(n+1,x) = (x+n+1)*R(n,x)-x*R'(n,x). Cf. A132159. - Peter Bala, Oct 28 2011
A padded shifted version of this lower triangular matrix with zeros in the first column and row except for a one in the diagonal position is given by integral(t=0 to t=infinity) exp[-t(I-P)] = 1/(I-P) = I + P^2 + P^3 + ... where P is the infinitesimal generator matrix A218234 and I the identity matrix. The non-padded version is given by P replaced by A132440. - Tom Copeland, Oct 25 2012
From Peter Bala, Aug 28 2013: (Start)
The row polynomials R(n,x) form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = Sum_{k=0..n} binomial(n,k)*y^(n-k)*R(k,x).
Let P(n,x) = Product_{k=0..n-1} (x + k) denote the rising factorial polynomial sequence with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (6, 6, 3, 1) so P(3,x + 1) = (x + 1)*(x + 2)*(x + 3) = 6 + 6*x + 3*x*(x + 1) + x*(x + 1)*(x + 2). (End)
From Tom Copeland, Apr 21 & 26, and Aug 13 2014: (Start)
T-I = M = -A021009*A132440*A021009 with e.g.f. y*exp(x*y)/(1-y). Cf. A132440. Dividing the n-th row of M by n generates the (n-1)th row of T.
T = 1/(I - A132440) = {2*I - exp[(A238385-I)]}^(-1) = unsigned exp[(I-A238385)] = exp[A000670(.)*(A238385-I)] = , umbrally, where I = identity matrix.
The e.g.f. is exp(x*y)/(1-y), so the row polynomials form an Appell sequence with lowering operator d/dx and raising operator x + 1/(1-D).
With L(n,m,x)= Laguerre polynomials of order m, the row polynomials are (-1)^n*n!*L(n,-1-n,x) = (-1)^n*(-1!/(-1-n)!)*K(-n,-1-n+1,x) = n!* K(-n,-n,x) where K is Kummer's confluent hypergeometric function (as a limit of n+s as s tends to zero).
Operationally, (-1)^n*n!*L(n,-1-n,-:xD:) = (-1)^n*x^(n+1)*:Dx:^n*x^(-1-n) = (-1)^n*x*:xD:^n*x^(-1) = (-1)^n*n!*binomial(xD-1,n) = n!*K(-n,-n,-:xD:) where :AB:^n = A^n*B^n for any two operators. Cf. A235706 and A132159.
The n-th row of signed M has the coefficients of d[(-:xD:)^n]/d(:Dx:)= f[d/d(-:xD:)](-:xD:)^n with f(y)=y/(y-1), :Dx:^n= n!L(n,0,-:xD:), and (-:xD:)^n = n!L(n,0,:Dx:). M has the coefficients of [D/(1-D)]x^n. (End)
From Tom Copeland, Nov 18 2015: (Start)
Coefficients of the row polynomials of the e.g.f. Sum_{n>=0} P_n(b1,b2,..,bn;t) x^n/n! = e^(P.(..;t) x) = e^(xt) / (1-b.x) = (1 + b1 x + b2 x^2 + b3 x^3 + ...) e^(xt) = 1 + (b1 + t) x + (2 b2 + 2 b1 t + t^2) x^2/2! + (6 b3 + 6 b2 t + 3 b1 t^2 + t^3) x^3/3! + ... , with lowering operator L = d/dt, i.e., L P_n(..;t) = n * P_(n-1)(..;t), and raising operator R = t + d[log(1 + b1 D + b2 D^2 + ...)]/dD = t - Sum_{n>=1} F(n,b1,..,bn) D^(n-1), i.e., R P_n(..,;t) = P_(n+1)(..;t), where D = d/dt and F(n,b1,..,bn) are the Faber polynomials of A263916.
Also P_n(b1,..,bn;t) = CIP_n(t-F(1,b1),-F(2,b1,b2),..,-F(n,b1,..,bn)), the cycle index polynomials A036039.
(End)
The raising operator R = x + 1/(1-D) = x + 1 + D + D^2 + ... in matrix form acting on an o.g.f. (formal power series) is the transpose of the production matrix M below. The linear term x is the diagonal of ones after transposition. The other transposed diagonals come from D^m x^n = n! / (n-m)! x^(n-m). Then P(n,x) = (1,x,x^2,..) M^n (1,0,0,..)^T is a matrix representation of R P(n-1,x) = P(n,x). - Tom Copeland, Aug 17 2016
The row polynomials have e.g.f. e^(xt)/(1-t) = exp(t*q.(x)), umbrally. With p_n(x) the row polynomials of A132013, q_n(x) = v_n(p.(u.(x))), umbrally, where u_n(x) = (-1)^n v_n(-x) = (-1)^n Lah_n(x), the Lah polynomials with e.g.f. exp[x*t/(t-1)]. This has the matrix form [T] = [q] = [v]*[p]*[u]. Conversely, p_n(x) = u_n (q.(v.(x))). - Tom Copeland, Nov 10 2016
From the Appell sequence formalism, 1/(1-b.D) t^n = P_n(b1,b2,..,bn;t), the generalized row polynomials noted in the Nov 18 2015 formulas, consistent with the 2007 comments. - Tom Copeland, Nov 22 2016
From Peter Bala, Feb 18 2017: (Start)
G.f.: Sum_{n >= 1} (n*x)^(n-1)/(1 + (n - t)*x)^n = 1 + (1 + t)*x + (2 + 2*t + t^2)*x^2 + ....
n-th row polynomial R(n,t) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(x + k)^k*(x + k - t)^(n-k) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(x + k)^(n-k)*(x + k + t)^k, for arbitrary x. The particular case of the latter sum when x = 0 and t = 1 is identity 10.35 in Gould, Vol.4. (End)
Rodrigues-type formula for the row polynomials: R(n, x) = -exp(x)*Int(exp(-x)* x^n, x), for n >= 0. Recurrence: R(n, x) =  x^n + n*R(n-1, x), for  n >= 1, and R(0, x) = 1. d/dx(R(n, x)) = R(n, x) - x^n, for n >= 0 (compare with the formula from Peter Bala, Aug 28 2013). - Wolfdieter Lang, Dec 23 2019
T(n, k) = Sum_{i=0..n-k} A048994(n-k, i) * n^i for 0 <= k <= n. - Werner Schulte, Jul 26 2022

Edited by Johannes W. Meijer, Oct 07 2009

New description from Dennis P. Walsh, Jan 24 2011

A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 6, 5, 5, 2, 3, 3, 5, 3, 5, 6, 7, 3, 6, 5, 9, 5, 9, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 10, 9, 9, 3, 6, 5, 9, 4, 9, 10, 12, 5, 9, 7, 13, 7, 12, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 6, 10, 9, 9, 3, 5, 6, 8, 5

Offset: 0

Gus Wiseman, Jun 14 2020

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. 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).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The a(n) patterns for n = 0, 1, 3, 7, 11, 13, 23, 83, 27, 45:
  0:  1:   11:   111:   211:   121:   2111:   2311:   1211:   2121:
---------------------------------------------------------------------
  ()  ()   ()    ()     ()     ()     ()      ()      ()      ()
      (1)  (1)   (1)    (1)    (1)    (1)     (1)     (1)     (1)
           (11)  (11)   (11)   (11)   (11)    (11)    (11)    (11)
                 (111)  (21)   (12)   (21)    (12)    (12)    (12)
                        (211)  (21)   (111)   (21)    (21)    (21)
                               (121)  (211)   (211)   (111)   (121)
                                      (2111)  (231)   (121)   (211)
                                              (2311)  (211)   (212)
                                                      (1211)  (221)
                                                              (2121)

John Tyler Rascoe, Table of n, a(n) for n = 0..8192
Wikipedia, Permutation pattern
Gus Wiseman, Statistics, classes, and transformations of standard compositions
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

References found in the links are not all included here.
Summing over indices with binary length n gives A335456(n).
The contiguous case is A335458.
The version for Heinz numbers of partitions is A335549.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Cf. A034691, A056986, A108917, A124767, A124770, A158005, A269134, A333218, A333222, A333224, A334030.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@Subsets[stc[n]]]],{n,0,30}]

from itertools import combinations
def comp(n):
    # see A357625
    return
def A335465(n):
    A,B,C = set(),set(),comp(n)
    c = range(len(C))
    for j in c:
        for k in combinations(c, j):
            A.add(tuple(C[i] for i in k))
    for i in A:
        D = {v: rank + 1 for rank, v in enumerate(sorted(set(i)))}
        B.add(tuple(D[v] for v in i))
    return len(B)+1 # John Tyler Rascoe, Mar 12 2025

A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).

Original entry on oeis.org

0, 1, 1, 3, 9, 31, 113, 431, 1697, 6847, 28161, 117631, 497665, 2128127, 9183489, 39940863, 174897665, 770452479, 3411959809, 15181264895, 67833868289, 304256253951, 1369404661761, 6182858317823, 27995941060609, 127100310290431, 578433619525633, 2638370120138751

Offset: 0

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

A simple context-free grammar.
Number of lattice paths from (0,0) to (2n-2,0) that stay (weakly) in the first quadrant and such that each step is either U=(1,1), D=(1,-1), or L=(3,1). Equivalently, underdiagonal lattice paths from (0,0) to (n-1,n-1) and such that each step is either (1,0), (0,1), or (2,1). E.g., a(4)=9 because in addition to the five Dyck paths from (0,0) to (6,0) [UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD] we have LDUD, LUDD, ULDD and UDLD. - Emeric Deutsch, Dec 21 2003
Hankel transform of a(n+1) is A006125(n+1). - Paul Barry, Apr 01 2007
Also, a(n+1) is the number of walks from (0,0) to (n,0) using steps (1,1), (1,-1) and (0,-1). See the U(n,k) array in A071943, where A052709(n+1) = U(n,0). - N. J. A. Sloane, Mar 29 2013
Diagonal sums of triangle in A085880. - Philippe Deléham, Nov 15 2013
From Gus Wiseman, Jun 17 2021: (Start)
Conjecture: For n > 0, also the number of sequences of length n - 1 covering an initial interval of positive integers and avoiding three terms (..., x, ..., y, ..., z, ...) such that x <= y <= z. The version avoiding the strict pattern (1,2,3) is A226316. Sequences covering an initial interval are counted by A000670. The a(1) = 1 through a(4) = 9 sequences are:
  ()  (1)  (1,1)  (1,2,1)
           (1,2)  (1,3,2)
           (2,1)  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (2,2,1)
                  (2,3,1)
                  (3,1,2)
                  (3,2,1)
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..499
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara, and Michael D. Weiner, Enumeration of colored Dyck paths via partial Bell polynomials, arXiv:1602.03550 [math.CO], 2016.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
L. Ferrari, E. Pergola, R. Pinzani, and S. Rinaldi, Jumping succession rules and their generating functions, Discrete Math., 271 (2003), 29-50.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 664
J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - _N. J. A. Sloane_, Dec 27 2012
D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Diagonal entries of A071943 and A071945.
Cf. A000108, A000670, A025227, A052709, A056986, A071943, A085880.
Cf. A102726, A117434, A158005, A226316, A333217, A335479.

[0] cat [(&+[Binomial(n,k+1)*Binomial(2*k,n-1): k in [0..n-1]])/n: n in [1..30]]; // G. C. Greubel, May 30 2022

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

InverseSeries[Series[(y-y^2)/(1+y^2), {y, 0, 24}], x] (* then A(x)= y(x) *) (* Len Smiley, Apr 12 2000 *)
CoefficientList[Series[(1 -Sqrt[1 -4x -4x^2])/(2(1+x)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 12 2016 *)

a(n)=polcoeff((1-sqrt(1-4*x*(1+x+O(x^n))))/2/(1+x),n)

[sum(binomial(k, n-k-1)*catalan_number(k) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, May 30 2022

a(n) + a(n-1) = A025227(n).
a(n) = Sum_{k=0..floor((n-1)/2)} (2*n-2-2*k)!/(k!*(n-k)!*(n-1-2*k)!). - Emeric Deutsch, Nov 14 2001
D-finite with recurrence: n*a(n) = (3*n-6)*a(n-1) + (8*n-18)*a(n-2) + (4*n-12)*a(n-3), n>2. a(1)=a(2)=1.
a(n) = b(1)*a(n-1) + b(2)*a(n-2) + ... + b(n-1)*a(1) for n>1 where b(n)=A025227(n).
G.f.: A(x) = x/(1-(1+x)*A(x)). - Paul D. Hanna, Aug 16 2002
G.f.: A(x) = x/(1-z/(1-z/(1-z/(...)))) where z=x+x^2 (continued fraction). - Paul D. Hanna, Aug 16 2002; revised by Joerg Arndt, Mar 18 2011
a(n+1) = Sum_{k=0..n} Catalan(k)*binomial(k, n-k). - Paul Barry, Feb 22 2005
From Paul Barry, Mar 14 2006: (Start)
G.f. is x*c(x*(1+x)) where c(x) is the g.f. of A000108.
Row sums of A117434. (End)
a(n+1) = (1/(2*Pi))*Integral_{x=2-2*sqrt(2)..2+2*sqrt(2)} x^n*(4+4x-x^2)/(2*(1+x)). - Paul Barry, Apr 01 2007
From Gary W. Adamson, Jul 22 2011: (Start)
For n>0, a(n) is the upper left term in M^(n-1), where M is an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  2, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  2, 2, 2, 1, 1, 0, ...
  2, 2, 2, 2, 1, 1, ...
  ... (End)
G.f.: x*Q(0), where Q(k) = 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
a(n) ~ sqrt(2-sqrt(2))*2^(n-1/2)*(1+sqrt(2))^(n-1)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jun 29 2013
a(n+1) = Sum_{k=0..floor(n/2)} A085880(n-k,k). - Philippe Deléham, Nov 15 2013

Better g.f. and recurrence from Michael Somos, Aug 03 2000

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A006154 Number of labeled ordered partitions of an n-set into odd parts.

Original entry on oeis.org

1, 1, 2, 7, 32, 181, 1232, 9787, 88832, 907081, 10291712, 128445967, 1748805632, 25794366781, 409725396992, 6973071372547, 126585529106432, 2441591202059281, 49863806091395072, 1074927056650469527, 24392086908129247232, 581176736647853024581

Offset: 0

Simon Plouffe

Conjecture: taking the sequence modulo an integer k gives an eventually periodic sequence. For example, the sequence taken modulo 10 is [1, 1, 2, 7, 2, 1, 2, 7, 2, 1, 2, 7, 2, ...], with an apparent period [1, 2, 7, 2] beginning at a(1), of length 4. Cf. A000670. - Peter Bala, Apr 12 2023

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

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Farid Aliniaeifard and Shu Xiao Li, Peak algebra in noncommuting variables, arXiv:2506.12868 [math.CO], 2025. See pp. 2, 9, 45.
Philippe Flajolet, Symbolic Enumerative Combinatorics and Complex Asymptotic Analysis, Algorithms Seminar 2000-2001, F. Chyzak (ed.), INRIA, (2002), pp. 161-170.
S. Getu and L. W. Shapiro, Combinatorial view of the composition of functions, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 8.

Cf. A000045, A000111, A000670, A003724, A193474, A196776.

readlib(coeftayl):
with(combinat, bell);
A:=series(1/(1-sinh(x)),x,20);
G(x):=A : f[0]:=G(x): for n from 0 to 21 do f[n]:=coeftayl(G(x), x=0, n);;
p[n]:=f[n]*((n)!) od: x:=0:seq(p[n], n=0..20); # Sergei N. Gladkovskii, Jun 01 2012
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((i->
      a(n-i)*binomial(n, i))(2*j+1), j=0..(n-1)/2))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 01 2022

a[n_] := Sum[ (-1)^i*(k - 2*i)^n*Binomial[k, i]/2^k, {k, 1, n}, {i, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Dec 07 2011, after Vladimir Kruchinin *)
With[{nn=20},CoefficientList[Series[1/(1-Sinh[x]),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Nov 16 2012 *)

a(n):=sum(sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k)/2^k,k,1,n); /* Vladimir Kruchinin, May 28 2011 */

a(n)=if(n<2,n>=0,sum(k=1,ceil(n/2),binomial(n,2*k-1)*a(n-2*k+1))) \\ Ralf Stephan

E.g.f.: 1/(1 - sinh(x)).
With alternating signs, e.g.f.: 1/(1+sinh(x)). - Ralf Stephan, Apr 29 2004
a(0) = a(1) = 1, a(n) = Sum_{k=1..ceiling(n/2)} C(n,2*k-1)*a(n-2*k+1). - Ralf Stephan, Apr 29 2004
a(n) ~ (sqrt(2)/2)*n!/log(1+sqrt(2))^(n+1). - Conjectured by Simon Plouffe, Feb 17 2007.
From Andrew Hone, Feb 22 2007: (Start)
This formula can be proved using the techniques in the article by Philippe Flajolet (see links) [see Theorem 5 and Table 2, noting that 1/(1-sinh(x)) just has a simple pole at x=log(1+sqrt(2))]. (End)
a(n) = Sum_{k=1..n} Sum_{i=0..k} (-1)^i*(k-2*i)^n*binomial(k,i)/2^k, n > 0, a(0)=1. - Vladimir Kruchinin, May 28 2011
Row sums (apart from a(0)) of A196776. - Peter Bala, Oct 06 2011
Row sums of A193474. - Peter Luschny, Oct 07 2011
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A003724 and A000111. - Peter Bala, Dec 06 2011
From Sergei N. Gladkovskii, Jun 01 2012: (Start)
Let E(x) be the e.g.f., then
E(x) = -1/x + 1/(x*(1-x))+ x^3/((1-x)*((1-x)*G(0) - x^2)); G(k) = (2*k+2)*(2*k+3)+x^2-(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction).
E(x) = -1/x + 1/(x*(1-x))+ x^3/((1-x)*((1-x)*G(0) - x^2)); G(k) = 8*k+6+x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction).
E(x) = 1/(1 - x*G(0)); G(k) = 1 + x^2/(2*(2*k+1)*(4*k+3) + 2*x^2*(2*k+1)*(4*k+3)/(-x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction).
(End).
E.g.f. 1/(1 - x*G(0)) where G(k) = 1 - x^2/( (2*k+1)*(2*k+3) - (2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 01 2012
O.g.f A(x) satisfies A(x) = 1 + ( A(x/(1-x))/(1-x) - A(x/(1+x))/(1+x) )/2. - Paul D. Hanna, Aug 19 2024

More terms from Christian G. Bower, Oct 15 1999

A026898 a(n) = Sum_{k=0..n} (n-k+1)^k.

Original entry on oeis.org

1, 2, 4, 9, 23, 66, 210, 733, 2781, 11378, 49864, 232769, 1151915, 6018786, 33087206, 190780213, 1150653921, 7241710930, 47454745804, 323154696185, 2282779990495, 16700904488706, 126356632390298, 987303454928973, 7957133905608837, 66071772829247410

Offset: 0

N. J. A. Sloane

nonn

Row sums of A004248, A009998, A009999.
First differences are in A047970.
First differences of A103439.
Antidiagonal sums of array A003992.
a(n-1), for n>=1, is the number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=1+max(prefix) for k>=1, that are simultaneously projections as maps f: [n] -> [n] where f(x)<=x and f(f(x))=f(x); see example and the two comments (Arndt, Apr 30 2011 Jan 04 2013) in A000110. - Joerg Arndt, Mar 07 2015
Number of finite sequences s of length n+1 whose discriminator sequence is s itself. Here the discriminator sequence of s is the one where the n-th term (n>=1) is the least positive integer k such that the first n terms are pairwise incongruent, modulo k. - Jeffrey Shallit, May 17 2016
From Gus Wiseman, Jan 08 2019: (Start)
Also the number of set partitions of {1,...,n+1} whose minima form an initial interval of positive integers. For example, the a(3) = 9 set partitions are:
  {{1},{2},{3},{4}}
  {{1},{2},{3,4}}
  {{1},{2,4},{3}}
  {{1,4},{2},{3}}
  {{1},{2,3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1,3,4},{2}}
  {{1,2,3,4}}
Missing from this list are:
  {{1},{2,3},{4}}
  {{1,2},{3},{4}}
  {{1,3},{2},{4}}
  {{1,2},{3,4}}
  {{1,2,3},{4}}
  {{1,2,4},{3}}
(End)
a(n) is the number of m-tuples of nonnegative integers less than or equal to n-m (including the "0-tuple"). - Mathew Englander, Apr 11 2021

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 23*x^4 + 66*x^5 + 210*x^6 + ...
where we have the identity:
A(x) = 1/(1-x) + x/(1-2*x) + x^2/(1-3*x) + x^3/(1-4*x) + x^4/(1-5*x) + ...
is equal to
A(x) = 1/(1-x) + x/((1-x)^2*(1+x)) + 2!*x^2/((1-x)^3*(1+x)*(1+2*x)) + 3!*x^3/((1-x)^4*(1+x)*(1+2*x)*(1+3*x)) + 4!*x^4/((1-x)^5*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...
From _Joerg Arndt_, Mar 07 2015: (Start)
The a(5-1) = 23 RGS described in the comment are (dots denote zeros):
01:  [ . . . . . ]
02:  [ . 1 . . . ]
03:  [ . 1 . . 1 ]
04:  [ . 1 . 1 . ]
05:  [ . 1 . 1 1 ]
06:  [ . 1 1 . . ]
07:  [ . 1 1 . 1 ]
08:  [ . 1 1 1 . ]
09:  [ . 1 1 1 1 ]
10:  [ . 1 2 . . ]
11:  [ . 1 2 . 1 ]
12:  [ . 1 2 . 2 ]
13:  [ . 1 2 1 . ]
14:  [ . 1 2 1 1 ]
15:  [ . 1 2 1 2 ]
16:  [ . 1 2 2 . ]
17:  [ . 1 2 2 1 ]
18:  [ . 1 2 2 2 ]
19:  [ . 1 2 3 . ]
20:  [ . 1 2 3 1 ]
21:  [ . 1 2 3 2 ]
22:  [ . 1 2 3 3 ]
23:  [ . 1 2 3 4 ]
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..500
Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, Set Partitions and Other Bell Number Enumerated Objects, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.
Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 12.
Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See p. 25.
Sajed Haque, Discriminators of Integer Sequences, 2017, See p. 33 Corollary 29.
Mathematics Stack Exchange, Asymptotics of ..., 2011.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019. See p. 3.
Robin D. P. Zhou, Pattern avoidance in revised ascent sequences, arXiv:2505.05171 [math.CO], 2025. See p. 4.

Cf. A000110, A000258, A000670, A003101, A008277, A038125, A062810.

Cf. A105795, A179928, A287215, A287216.

a026898 n = sum $ zipWith (^) [n + 1, n .. 1] [0 ..]
-- Reinhard Zumkeller, Sep 14 2014

[(&+[(n-k+1)^k: k in [0..n]]): n in [0..50]]; // Stefano Spezia, Jan 09 2019

a:= n-> add((n+1-j)^j, j=0..n): seq(a(n), n=0..23); # Zerinvary Lajos, Apr 18 2009

Table[Sum[(n-k+1)^k, {k,0,n}], {n, 0, 25}] (* Michael De Vlieger, Apr 01 2015 *)

{a(n)=polcoeff(sum(m=0,n,x^m/(1-(m+1)*x+x*O(x^n))),n)} /* Paul D. Hanna, Sep 13 2011 */

{INTEGRATE(n,F)=local(G=F);for(i=1,n,G=intformal(G));G}
{a(n)=local(A=1+x);A=sum(k=0,n,INTEGRATE(k,exp((k+1)*x+x*O(x^n))));n!*polcoeff(A,n)} \\ Paul D. Hanna, Dec 28 2013
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff( sum(m=0, n, m!*x^m/(1-x +x*O(x^n))^(m+1)/prod(k=1, m, 1+k*x +x*O(x^n))), n)}  /* From o.g.f. (Paul D. Hanna, Jul 20 2014) */
for(n=0, 25, print1(a(n), ", "))

[sum((n-j+1)^j for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 15 2021

a(n) = A003101(n) + 1.
G.f.: Sum_{n>=0} x^n/(1 - (n+1)*x). - Paul D. Hanna, Sep 13 2011
G.f.: G(0) where G(k) = 1 + x*(2*k*x-1)/((2*k*x+x-1) - x*(2*k*x+x-1)^2/(x*(2*k*x+x-1) + (2*k*x+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
E.g.f.: Sum_{n>=0} Integral^n exp((n+1)*x) dx^n, where Integral^n F(x) dx^n is the n-th integration of F(x) with no constant of integration. - Paul D. Hanna, Dec 28 2013
O.g.f.: Sum_{n>=0} n! * x^n/(1-x)^(n+1) / Product_{k=1..n} (1 + k*x). - Paul D. Hanna, Jul 20 2014
a(n) = A101494(n+1,0). - Vladimir Kruchinin, Apr 01 2015
a(n-1) = Sum_{k = 1..n} k^(n-k). - Gus Wiseman, Jan 08 2019
log(a(n)) ~ (1 - 1/LambertW(exp(1)*n)) * n * log(1 + n/LambertW(exp(1)*n)). - Vaclav Kotesovec, Jun 15 2021
a(n) ~ sqrt(2*Pi/(n+1 + (n+1)/w(n))) * ((n+1)/w(n))^(n+2 - (n+1)/w(n)), where w(n) = LambertW(exp(1)*(n+1)). - Vaclav Kotesovec, Jun 25 2021, after user "leonbloy", see Mathematics Stack Exchange link.

a(23)-a(25) from Paul D. Hanna, Dec 28 2013

A328509 Number of non-unimodal sequences of length n covering an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 3, 41, 425, 4287, 45941, 541219, 7071501, 102193755, 1622448861, 28090940363, 526856206877, 10641335658891, 230283166014653, 5315654596751659, 130370766738143517, 3385534662263335179, 92801587315936355325, 2677687796232803000171, 81124824998464533181661

Offset: 0

Gus Wiseman, Feb 19 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(3) = 3 sequences are (2,1,2), (2,1,3), (3,1,2).
The a(4) = 41 sequences:
  (1212)  (2113)  (2134)  (2413)  (3142)  (3412)
  (1213)  (2121)  (2143)  (3112)  (3212)  (4123)
  (1312)  (2122)  (2212)  (3121)  (3213)  (4132)
  (1323)  (2123)  (2213)  (3122)  (3214)  (4213)
  (1324)  (2131)  (2312)  (3123)  (3231)  (4231)
  (1423)  (2132)  (2313)  (3124)  (3241)  (4312)
  (2112)  (2133)  (2314)  (3132)  (3312)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Not requiring non-unimodality gives A000670.
The complement is counted by A007052.
The case where the negation is not unimodal either is A332873.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Unimodal compositions covering an initial interval are A227038.
Numbers whose unsorted prime signature is not unimodal are A332282.
Covering partitions with unimodal run-lengths are A332577.
Non-unimodal compositions covering an initial interval are A332743.
Cf. A060223, A255906, A332281, A332284, A332639, A332672, A332834, A332870.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Union@@Permutations/@allnorm[n],!unimodQ[#]&]],{n,0,5}]

seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - (1 - 3*x + x^2)/(1 - 4*x + 2*x^2), -(n+1)) \\ Andrew Howroyd, Jan 28 2024

a(n) = A000670(n) - A007052(n-1) for n > 0. - Andrew Howroyd, Jan 28 2024

a(9) from Robert Price, Jun 19 2021

a(10) onwards from Andrew Howroyd, Jan 28 2024

A262809 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one or more components by one; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 13, 1, 1, 1, 75, 409, 63, 1, 1, 1, 541, 23917, 16081, 321, 1, 1, 1, 4683, 2244361, 10681263, 699121, 1683, 1, 1, 1, 47293, 308682013, 14638956721, 5552351121, 32193253, 8989, 1, 1, 1, 545835, 58514835289, 35941784497263, 117029959485121, 3147728203035, 1538743249, 48639, 1, 1

Offset: 0

Alois P. Heinz, Oct 02 2015

Also, A(n,k) is the number of alignments for k sequences of length n each (Slowinski 1998).
Row r > 0 is asymptotic to sqrt(r*Pi) * (r^(r-1)/(r-1)!)^n * n^(r*n+1/2) / (2^(r/2) * exp(r*n) * (log(2))^(r*n+1)), or equivalently to sqrt(r) * (r^(r-1)/(r-1)!)^n * (n!)^r / (2^r * (Pi*n)^((r-1)/2) * (log(2))^(r*n+1)). - Vaclav Kotesovec, Mar 23 2016
From Vaclav Kotesovec, Mar 23 2016: (Start)
Column k > 0 is asymptotic to sqrt(c(k)) * d(k)^n / (Pi*n)^((k-1)/2), where c(k) and d(k) are roots of polynomial equations of degree k, independent on n.
---------------------------------------------------
k               d(k)
---------------------------------------------------
2              5.8284271247461900976033774484193...
3             56.9476283720414911685286267804411...
4            780.2794068067951456595241495989622...
5          13755.2719024115081712083954421541320...
6         296476.9162644200814909862281498491264...
7        7553550.6198338218721069097516499501996...
8      222082591.6017202421029000117685530884167...
9     7400694480.0494436216324852038000444393262...
10  275651917450.6709238286995776605620357737005...
---------------------------------------------------
d(k) is a root of polynomial:
---------------------------------------------------
k=2, 1 - 6*d + d^2
k=3, -1 + 3*d - 57*d^2 + d^3
k=4, 1 - 12*d - 218*d^2 - 780*d^3 + d^4
k=5, -1 + 5*d - 1260*d^2 - 3740*d^3 - 13755*d^4 + d^5
k=6, 1 - 18*d - 5397*d^2 - 123696*d^3 + 321303*d^4 - 296478*d^5 + d^6
k=7, -1 + 7*d - 24031*d^2 - 374521*d^3 - 24850385*d^4 + 17978709*d^5 - 7553553*d^6 + d^7
k=8, 1 - 24*d - 102692*d^2 - 9298344*d^3 + 536208070*d^4 - 7106080680*d^5 - 1688209700*d^6 - 222082584*d^7 + d^8
(End)
d(k) = (2^(1/k) - 1)^(-k). - David Bevan, Apr 07 2022
d(k) is asymptotic to (k/log(2))^k/sqrt(2). - David Bevan, Apr 07 2022
A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column. - Andrew Howroyd, Jan 23 2020

			A(2,2) = 13: [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,1),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)].
Square array A(n,k) begins:
  1, 1,    1,        1,             1,                   1, ...
  1, 1,    3,       13,            75,                 541, ...
  1, 1,   13,      409,         23917,             2244361, ...
  1, 1,   63,    16081,      10681263,         14638956721, ...
  1, 1,  321,   699121,    5552351121,     117029959485121, ...
  1, 1, 1683, 32193253, 3147728203035, 1050740615666453461, ...

Alois P. Heinz, Antidiagonals n = 0..48, flattened
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Columns: A000012 (k=0 and k=1), A001850 (k=2), A126086 (k=3), A263064 (k=4), A263065 (k=5), A263066 (k=6), A263067 (k=7), A263068 (k=8), A263069 (k=9), A263070 (k=10).
Rows: A000012 (n=0), A000670 (n=1), A055203 (n=2), A062208 (n=3), A062205 (n=4), A263061 (n=5), A263062 (n=6), A062204 (n=7), A263063 (n=8), A263071 (n=9), A263072 (n=10).
Main diagonal: A262810.
Cf. A210472, A225094, A227578, A227655, A229142, A229345, A263159, A316674.
Cf. A188392, A330942, A331461, A331637.

A:= (n, k)-> add(add((-1)^i*binomial(j, i)*
     binomial(j-i, n)^k, i=0..j), j=0..k*n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[, 0] =  1; A[n, k_] := Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jul 22 2016, after Alois P. Heinz *)

T(n,k) = {my(m=n*k); sum(j=0, m, binomial(j,n)^k*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))} \\ Andrew Howroyd, Jan 23 2020

A(n,k) = Sum_{j=0..k*n} Sum_{i=0..j} (-1)^i*C(j,i)*C(j-i,n)^k.
A(n,k) = Sum_{i >= 0} binomial(i,n)^k/2^(i+1). - Peter Bala, Jan 30 2018
A(n,k) = Sum_{j=0..n*k} binomial(j,n)^k * Sum_{i=j..n*k} (-1)^(i-j) * binomial(i,j). - Andrew Howroyd, Jan 23 2020

A036035 Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030, 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510, 256, 384, 576, 864, 1296, 960, 1440, 2160

Offset: 0

N. J. A. Sloane

The exponents can be read off Abramowitz and Stegun, p. 831, column labeled "pi".
Here are the partitions in the order used by Abramowitz and Stegun (graded reflected colexicographic order): 0; 1; 2, 1+1; 3, 1+2, 1+1+1; 4, 1+3, 2+2, 1+1+2, 1+1+1+1; 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1; ... (Cf. A036036)
Here are the partitions in graded colexicographic order: 0; 1; 2, 1+1; 3, 2+1, 1+1+1; 4, 3+1, 2+2, 2+1+1, 1+1+1+1; 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; ... (Cf. A036037)
Since the prime signature is a partition of Omega(n), so to speak, the internal order is only a matter of convention and has no effect on the least integer with a given prime signature.
The graded colexicographic order has the advantage that the exponents are in the same order as the least integer with a given prime signature (also used on the wiki page, see links).
Embedded values include the primorial numbers 1, 2, 6, 30, 210, 2310, 30030 ... (A002110) with unordered factorizations counted by A000110 (Bell numbers) and ordered factorizations by A000670 (ordered Bell numbers).
When viewed as a table the n-th row has partition(n) (A000041(n)) terms. - Alford Arnold, Jul 31 2003
A closely related sequence, A096443(n), gives the number of partitions of the n-th multiset. - Alford Arnold, Sep 29 2005

			1;
2;
4, 6;
8, 12, 30;
16, 24, 36, 60, 210;
32, 48, 72, 120, 180, 420, 2310;
64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030;
128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510;

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Peter Luschny, Rows n = 0..25, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
John Baez, What happens when a particle gets created?
OEIS Wiki, Prime signature.

A025487 in a different order. Cf. A035098, A002110, A000110 and A000670.

Cf. A025487, A059901, A096443.

with(combinat):
A036035_row := proc(n) local e, w; w := proc(e) local i, p;
p := [seq(ithprime(nops(e)-i+1), i=1..nops(e))];
mul(p[i]^e[i], i=1..nops(e)) end:
seq(w(conjpart(e)), e = partition(n)) end:
seq(A036035_row(i), i=0..10);  # Peter Luschny, Aug 01 2013

nmax = 52; primeSignature[n_] := Sort[ FactorInteger[n], #1[[2]] > #2[[2]] & ][[All, 2]]; ip[n_] := Reverse[ Sort[#]] & /@ Split[ Sort[ IntegerPartitions[n], Length[#1] < Length[#2] & ], Length[#1] == Length[#2] & ]; tip = Flatten[ Table[ip[n], {n, 0, 8}], 2]; a[n_] := (sig = tip[[n+1]]; k = 1; While[sig =!= primeSignature[k++]]; k-1); a[0] = 1; a[1] = 2; Table[an = a[n]; Print[an]; an, {n, 0, nmax}](* Jean-François Alcover, Nov 16 2011 *)

Row(n)={[prod(i=1, #p, prime(i)^p[#p+1-i]) | p<-partitions(n)]} \\ Andrew Howroyd, Oct 19 2020

More terms from Alford Arnold; corrected Sep 10 2002
More terms from Ray Chandler, Jul 13 2003
Definition corrected by Daniel Forgues, Jan 16 2011

A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038.

Original entry on oeis.org

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

Offset: 0

Alford Arnold

This array gives in row n>=1 the multinomial numbers (call them M_0 numbers) m!/product((a_j)!,j=1..n) with the exponents of the partitions of n with number of parts m:=sum(a_j,j=1..n), given in the Abramowitz-Stegun order. See p. 831 of the given reference. See also the arrays for the M_1, M_2 and M_3 multinomial numbers A036038, A036039 and A036040 (or A080575).
For a signed version see A111786.
These M_0 multinomial numbers give the number of compositions of n >= 1 with parts corresponding to the partitions of n (in A-St order). See an n = 5 example below. The triangle with the summed entries of like number of parts m is A007318(n-1, m-1) (Pascal). - Wolfdieter Lang, Jan 29 2021

			Table starts:
[1]
[1]
[1, 1]
[1, 2, 1]
[1, 2, 1, 3, 1]
[1, 2, 2, 3, 3, 4, 1]
[1, 2, 2, 1, 3, 6, 1, 4, 6,  5, 1]
[1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1]
.
T(5,6) = 4 because there are four multisets using the first four digits {0,1,2,3}: 32100, 32110, 32210 and 33210
T(5,6) = 4 because there are 4 compositions of 5 that can be formed from the partition 2+1+1+1. - _Geoffrey Critzer_, May 19 2013
These 4 compositions 2+1+1+1, 1+2+1+1, 1+1+2+1 and 1+1+1+2 of 5 correspond to the 4 set partitions of [5] :={1,2,3,4,5}, with 4 blocks of consecutive numbers, namely {1,2},{3},{4},{5} and {1},{2,3},{4},{5} and {1},{2},{3,4},{5} and {1},{2},{3},{4,5}. - _Wolfdieter Lang_, May 30 2018

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972.
Wolfdieter Lang, First 10 rows and more.

Cf. A000670, A007318, A036035, A036038, A019538, A115621, A309004, A000079 (row sums), A000041 (row lengths).

nmax:=9: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036040(n, m) := (add(q(t), t=1..n))!/(mul(q(t)!, t=1..n)); od: od: seq(seq(A036040(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016

C(sig)={my(S=Set(sig)); (#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

from collections import Counter
def ASPartitions(n, k):
    Q = [p.to_list() for p in Partitions(n, length=k)]
    for q in Q: q.reverse()
    return sorted(Q)
def A048996_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [factorial(len(p))//h(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A048996_row(n)) # Peter Luschny, Nov 02 2019 [corrected on notice from Sean A. Irvine, Apr 30 2022]

T(n,k) = A036040(n,k) * Factorial(A036043(n,k)) / A036038(n,k) = A049019(n,k) / A036038(n,k).
If the n-th partition is P, a(n) is the multinomial coefficient of the signature of P. - Franklin T. Adams-Watters, May 30 2006
T(n,k) = A309004(A036035(n,k)). - Andrew Howroyd, Oct 19 2020

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

a(0)=1 prepended by Andrew Howroyd, Oct 19 2020

cp's OEIS Frontend

A019536 Number of length n necklaces with integer entries that cover an initial interval of positive integers.

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A094587 Triangle of permutation coefficients arranged with 1's on the diagonal. Also, triangle of permutations on n letters with exactly k+1 cycles and with the first k+1 letters in separate cycles.

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A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

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A052709 Expansion of g.f. (1-sqrt(1-4*x-4*x^2))/(2*(1+x)).

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A006154 Number of labeled ordered partitions of an n-set into odd parts.

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A026898 a(n) = Sum_{k=0..n} (n-k+1)^k.

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A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).