A187251 -id:A187251 - cp's OEIS frontend

A007405 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2.

1, 2, 6, 24, 116, 648, 4088, 28640, 219920, 1832224, 16430176, 157554048, 1606879040, 17350255744, 197553645440, 2363935624704, 29638547505408, 388328781668864, 5304452565517824, 75381218537805824, 1112348880749130752, 17014743624340539392, 269360902955086379008

Offset: 0

N. J. A. Sloane

Binomial transform of A004211.
Equals leftmost term in iterates of M^n * [1,1,1,...], where M = a bidiagonal matrix with (1,3,5,7,...) in the main diagonal and (1,1,1,...) in the superdiagonal. - Gary W. Adamson, Apr 13 2009
This is the number of type B set partitions, see R. Suter. - Per W. Alexandersson, Dec 19 2022

			a(4) = 116 = sum of top row terms of M^3 = (49 + 44 + 18 + 4 + 1).

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..100
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
Hasan Arslan, Nazmiye Alemdar, Mariam Zaarour, and Hüseyin Altındiş, On Bell numbers of type D, arXiv:2504.16522 [math.CO], 2025. See p. 3.
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 2.
Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 25.
Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, Flattened Stirling Permutations, arXiv:2306.13034 [math.CO], 2023. See p. 14.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
John M. Neuberger, Nándor Sieben, and James W. Swift, Invariant Polydiagonal Subspaces of Matrices and Constraint Programming, arXiv:2411.10904 [math.DS], 2024. See p. 7.
Tilman Piesk, Sets of fixed points of permutations of the n-cube: a(3)=24 for the cube and a(4)=116 for the tesseract.
N. J. A. Sloane, Transforms
R. Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Cf. A000110 (b=1), this sequence (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

Cf. A004211, A039755, A039756, A187251.

m:=20; c:=2; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019

max = 19; f[x_]:= Exp[x + Exp[2x]/2 -1/2]; CoefficientList[Series[f[x], {x,0,max}], x]*Range[0, max]! (* Jean-François Alcover, Nov 22 2011 *)
Table[Sum[Binomial[n, k] * 2^k * BellB[k, 1/2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

x='x+O('x^66); Vec(serlaplace(exp(x+1/2*exp(2*x)-1/2))) \\ Joerg Arndt, May 13 2013

@CachedFunction
def S(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return S(n-1, k-1, m) + (m*(k+1)-1)*S(n-1, k, m)
def A007405(n): return add(S(n, k, 2) for k in (0..n)) # Peter Luschny, May 20 2013

b=2;
def A007405_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(x +(exp(b*x)-1)/b) ).egf_to_ogf().list()
A007405_list(30) # G. C. Greubel, Feb 24 2019

E.g.f.: exp(x + (exp(2*x) - 1)/2).
Row sums of triangles A039755, A039756. - Philippe Deléham, Feb 20 2005
a(n) = sum of top row terms of M^n, M = an infinite square production matrix in which a diagonal of 1's is appended to the right of Pascal's triangle squared; as follows:
   1,  1,  0, 0, 0, 0, ...
   2,  1,  1, 0, 0, 0, ...
   4,  4,  1, 1, 0, 0, ...
   8, 12,  6, 1, 1, 0, ...
  16, 32, 24, 8, 1, 1, ...
   ... - Gary W. Adamson, Aug 01 2011
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(2*k+1)*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: -G(0) where G(k) = 1 - (x*(2*k+1) - 2)/(x*(2*k+1) - 1 - x*(x*(2*k+1) - 1)/(x + (x*(2*k+1) - 2)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 29 2013
G.f.: 1/Q(0), where Q(k) = 1 - 2*(k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/Q(0), where Q(k) = 1 - x - x/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 13 2013
G.f.: 1/(1-x*Q(0)), where Q(k) = 1 + x/(1 - x + 2*x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
Conjecture: Let M_n be an n X n matrix whose elements are m_ij = 1 for i < j - 1, m_ij = -1 for i = j - 1, and m_ij = binomial(n - i,j - i) otherwise. Then a(n - 1) = Det(M_n). - Benedict W. J. Irwin, Apr 19 2017
a(n) = exp(-1/2) * Sum_{k>=0} (2*k + 1)^n / (2^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) = Sum_{k=0..n} binomial(n,k) * A187251(k) * A187251(n-k). - Vaclav Kotesovec, Apr 17 2020
a(n) ~ 2^(n + 1/2) * n^(n + 1/2) * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^(n + 1/2)). - Vaclav Kotesovec, Jun 26 2022

Name edited by G. C. Greubel, Feb 24 2019

A344855 Number T(n,k) of permutations of [n] having k cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 4, 11, 6, 1, 0, 8, 40, 35, 10, 1, 0, 16, 148, 195, 85, 15, 1, 0, 32, 560, 1078, 665, 175, 21, 1, 0, 64, 2160, 5992, 5033, 1820, 322, 28, 1, 0, 128, 8448, 33632, 37632, 17913, 4284, 546, 36, 1, 0, 256, 33344, 190800, 280760, 171465, 52941, 9030, 870, 45, 1

Offset: 0

Alois P. Heinz, May 30 2021

The sequence of column k satisfies a linear recurrence with constant coefficients of order k*(k+1)/2 = A000217(k).

			T(4,1) = 4: (1234), (1243), (1423), (1432).
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    1;
  0,  2,    3,    1;
  0,  4,   11,    6,    1;
  0,  8,   40,   35,   10,    1;
  0, 16,  148,  195,   85,   15,   1;
  0, 32,  560, 1078,  665,  175,  21,  1;
  0, 64, 2160, 5992, 5033, 1820, 322, 28, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-10 give: A000007, A166444, A346317, A346318, A346319, A346320, A346321, A346322, A346323, A346324, A346325.
Row sums give A187251.
Main diagonal gives A000012, lower diagonal gives A000217, second lower diagonal gives A000914.
T(n+1,n) gives A000217.
T(n+2,n) gives A000914.
T(2n,n) gives A345342.
Cf. A060281, A132393, A186366, A345341.

b:= proc(n) option remember; `if`(n=0, 1, add(expand(x*
      b(n-j)*binomial(n-1, j-1)*ceil(2^(j-2))), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..12);

b[n_] := b[n] = If[n == 0, 1, Sum[Expand[x*b[n-j]*
     Binomial[n-1, j-1]*Ceiling[2^(j-2)]], {j, n}]];
T[n_] := CoefficientList[b[n], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A345341(n).

For fixed k, T(n,k) ~ (2*k)^n / (4^k * k!). - Vaclav Kotesovec, Jul 15 2021

A187250 Triangle read by rows: T(n,k) is the number of permutations of [n] having k cycles with at least 3 alternating runs (it is assumed that the smallest element of a cycle is in the first position), 0<=k<=floor(n/4).

Original entry on oeis.org

1, 1, 2, 6, 22, 2, 94, 26, 460, 260, 2532, 2508, 15420, 24760, 140, 102620, 254968, 5292, 739512, 2760432, 128856, 5729192, 31547344, 2640264, 47429896, 381339368, 50186136, 46200, 417429800, 4879612808, 926494712, 3483480, 3888426512, 66107044176, 17025751600, 157068912

Offset: 0

Emeric Deutsch, Mar 08 2011

Number of entries in row n is 1+floor(n/4).
Sum of entries in row n is n!.
T(n,0)=A187251(n).
Sum(k*T(n,k), k>=0) = A187252(n).

			T(4,1)=2 because we have (1324) and (1423).
Triangle starts:
1;
1;
2;
6;
22,2;
94,26;
460,260;

Cf. A187244, A187247, A187251, A187252.

G := exp((1/4*(1-t))*(2*z-1+exp(2*z)))/(1-z)^t: Gser := simplify(series(G, z = 0, 17)): for n from 0 to 14 do P[n] := sort(factorial(n)*coeff(Gser, z, n)) end do: for n from 0 to 14 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form

E.g.f.: G(t,z) = exp[(1/4)(1-t)(2z-1+exp(2z))]/(1-z)^t.
The 4-variate g.f. H(u,v,w,z) (exponential with respect z), where u marks number of cycles with 1 alternating run, v marks number of cycles with 2 alternating runs, w marks the number of all cycles, and z marks the size of the permutation, is given by H(u,v,w,z) = exp[(1/4)w((v-1)(exp(2z)+2z)+4(u-v)exp(z)+1-4u+3v)]/(1-z)^w.
We have G(t,z) = H(1/t,1/t,t,z).

A216964 Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points.

Original entry on oeis.org

1, 2, 6, 22, 2, 94, 26, 460, 244, 16, 2532, 2124, 384, 15420, 18536, 6092, 272, 102620, 166440, 83436, 10384, 739512, 1550864, 1082712, 247776, 7936, 5729192, 15040112, 13841928, 4864480, 441088, 47429896, 151960264, 177512632, 87003032, 14741984, 353792

Offset: 1

N. J. A. Sloane, Sep 27 2012

See Ma and Chow (2012) for precise definition (see Corollary 5).

			Triangle begins:
1
2
6
22, 2
94, 26
460, 244, 16
2532, 2124, 384
...

Shi-Mei Ma and Chak-On Chow, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012.

First column is A187251.

rows = 12;
Reap[For[P = x*y; n = 1; Sow[{1}], n < rows, n++, P = (n*q + x*y)*P + 2*q*(1-q)*D[P, q] + 2*x*(1-q)*D[P, x] + (1-2*y+q*y)*D[P, y] // Simplify; Sow[CoefficientList[P /. {x -> 1, y -> 1}, q]]]][[2, 1]] // Flatten (* Jean-François Alcover, Sep 23 2018, from PARI *)

tabf(m) = {P = x*y; for (n=1, m, M = subst(P, x, 1); M = subst(M, y, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); P = (n*q+x*y)*P + 2*q*(1-q)*deriv(P, q)+ 2*x*(1-q)*deriv(P,x)+ (1-2*y+q*y)*deriv(P,y););} \\ Michel Marcus, Feb 08 2013

More terms from Michel Marcus, Feb 08 2013

A347011 Euler transform of j-> ceiling(2^(j-2)).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 93, 207, 453, 999, 2185, 4796, 10470, 22871, 49815, 108427, 235515, 511074, 1107248, 2396299, 5179169, 11181877, 24113939, 51949572, 111801422, 240381703, 516355235, 1108186951, 2376314763, 5091422730, 10900063776, 23317805916

Offset: 0

Alois P. Heinz, Aug 10 2021

nonn

Differs from A206301 first at n=10.

Alois P. Heinz, Table of n, a(n) for n = 0..3250

Cf. A034691, A034899, A166861, A187251, A206301, A319918.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, i-1)*binomial(ceil(2^(i-2))+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
       ceil(2^(d-2)), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);

CoefficientList[Series[1/(1-x) * Product[1/(1 - x^k)^(2^(k-2)), {k, 2, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 11 2021 *)

G.f.: Product_{j>0} 1/(1-x^j)^ceiling(2^(j-2)).

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A007405 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2.

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A344855 Number T(n,k) of permutations of [n] having k cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A187250 Triangle read by rows: T(n,k) is the number of permutations of [n] having k cycles with at least 3 alternating runs (it is assumed that the smallest element of a cycle is in the first position), 0<=k<=floor(n/4).

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A216964 Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points.

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Programs

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Extensions

A347011 Euler transform of j-> ceiling(2^(j-2)).

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