A029767 -id:A029767 - cp's OEIS frontend

A053482 Binomial transform of A029767.

1, 4, 21, 142, 1201, 12336, 149989, 2113546, 33926337, 611660476, 12243073621, 269456124774, 6468249055921, 168191402251432, 4709596238204901, 141291441773619106, 4521383010795364609, 153727989225714801396, 5534225015581836134677

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

This is the column k=3 of an array T(n,k) = A181783(n,k) defined by T(n,0)=T(0,k)=1 and T(n,k) = n*(k-1)*T(n-1,k) +T(n,k-1), which starts
   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
   1,   1,   2,   4,   7,  11,  16,  22,  29,  37,  46,...
   1,   1,   5,  21,  63, 151, 311, 575, 981,1573,2401,...
   1,   1,  16, 142, 709,2521,7186,17536,38137,75889,140716,...
   1,   1,  65,1201,9709,50045,193765,614629,1682465,4110913,9176689,...
Column k=2 is A000522. The e.g.f. for column k is E_k(z) = E_(k-1)(z)/[1-(k-1)] = exp(z)/prod_{j=1..k-1} (1-j*z). - Richard Choulet, Dec 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[E^x/(1-3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

E.g.f.: exp(x)*(2/(1-2x)-1/(1-x))=exp(x)/(1-3x+2x^2); a(n)=sum{k=0..n, C(n,k)*k!*(2^(k+1)-1)}; a(n)=n!*sum{k=0..n, (2^(n-k+1)-1)/k!}; a(n)=int(x^n*(exp((1-x)/2)-exp(1-x)),x,1,infty); a(n)=2*A010844(n)-A000522(n); - Paul Barry, Jan 28 2008
Conjecture: a(n) -(3*n+1)*a(n-1) +(2*n+3)*(n-1)*a(n-2) -2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 29 2012
a(n) = 3*n*a(n-2)-2*n*(n-1)*a(n-2)+1, derived from the array defined in the comment, which proves the previous conjecture. - Richard Choulet, Dec 17 2012
a(n) ~ n! * 2^(n+1)*exp(1/2). - Vaclav Kotesovec, Oct 02 2013

A052840 a(n) = n*A029767(n-1).

Original entry on oeis.org

0, 0, 2, 9, 56, 450, 4464, 52920, 731520, 11566800, 206035200, 4083488640, 89137843200, 2124970848000, 54929029478400, 1530259226496000, 45705137084006400, 1456873475016960000, 49362677881380864000

Offset: 0

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

Old name was: A simple grammar.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 807

spec := [S,{B=Sequence(Z,1 <= card),C=Cycle(B),S=Prod(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# alternative
A052840 := proc(n)
    log((-1+x)/(-1+2*x))*x ;
    coeftayl(%,x=0,n)*n! ;
end proc:
seq(A052840(n),n=0..20) ; # R. J. Mathar, Jan 20 2025

Flatten[{0, 0, Table[n!*(2^(n-1) - 1)/(n-1), {n, 2, 20}]}] (* Vaclav Kotesovec, Jun 06 2019 *)

E.g.f.: log((-1+x)/(-1+2*x))*x.
D-finite with recurrence: a(1)=0, a(2)=2, (-2*n+2*n^3-4+4*n^2)*a(n)+(-6*n-3*n^2)*a(n+1)+(n+1)*a(n+2), i.e. (-n+1)*a(n) +3*n*(n-2)*a(n-1) -2*n*(n-1)*(n-3)*a(n-2)=0
For n > 1, a(n) = n! * (2^(n-1) - 1)/(n-1). - Vaclav Kotesovec, Jun 06 2019

A053481 First differences of A029767.

Original entry on oeis.org

2, 11, 76, 654, 6816, 83880, 1193760, 19318320, 350622720, 7056927360, 156031142400, 3760042809600, 98093779660800, 2754553785984000, 82841868639129600, 2656672553703168000, 90498598469959680000, 3263440333591646208000

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

a[n_] := (n - 1)! * (2^n - 1); Table[a[n+2] - a[n+1], {n,0,25}] (* G. C. Greubel, Jan 19 2017 *)

for(n=0,25, print1(( (n+1)!*(2^(n+2) -1) - n!*(2^(n+1) -1)), ", ")) \\ G. C. Greubel, Jan 19 2017

a(n) = (n+1)!*(2^(n+2) -1) - n!*(2^(n+1) -1) for n>=0. - G. C. Greubel, Jan 19 2017

A053483 Euler transform of A029767.

Original entry on oeis.org

1, 4, 18, 114, 900, 8845, 103861, 1427122, 22486706, 399906140, 7922936720, 173013117604, 4127746294408, 106806183646594, 2978731438384738, 89065499057526433, 2842061902985159593, 96395720127638538076, 3462922846509648162418

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A107895.

Rest[CoefficientList[Series[Product[1/(1 - x^k)^((k-1)!*(2^k-1)), {k, 1, 20}], {x, 0, 20}], x]] (* Vaclav Kotesovec, Aug 07 2015 *)

A131222 Exponential Riordan array [1, log((1-x)/(1-2x))].

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 14, 9, 1, 0, 90, 83, 18, 1, 0, 744, 870, 275, 30, 1, 0, 7560, 10474, 4275, 685, 45, 1, 0, 91440, 143892, 70924, 14805, 1435, 63, 1, 0, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1

Offset: 0

Paul Barry, Jun 18 2007

This is also the matrix product of the unsigned Lah numbers and the Stirling cycle numbers. See also A079639 and A079640 for variants based on an (1,1)-offset of the number triangles. - Peter Luschny, Apr 12 2015
The Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016
Essentially the same as A079638. - Peter Bala, Feb 12 2022

			Number triangle starts:
  1,
  0,   1;
  0,   3,   1;
  0,  14,   9,   1;
  0,  90,  83,  18,  1;
  0, 744, 870, 275, 30,  1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Paul Barry, Exponential Riordan arrays and permutation enumeration,Journal of Integer Sequences, Vol. 13 (2010).

Cf. A000007, A002866, A029767, A079638, A079639, A079640.

RioExp := (d,h,n,k) -> coeftayl(d*h^k, x=0,n)*n!/k!:
A131222 := (n,k) -> RioExp(1,log((1-x)/(1-2*x)),n,k):
seq(print(seq(A131222(n,k),k=0..n)),n=0..5); # Peter Luschny, Apr 15 2015
# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0,1,n!*(2^(n+1)-1)), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 1, #! (2^(#+1) - 1)]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

T(n,m):=if n=0 and m=0 then 1 else n!*sum((stirling1(k,m)*2^(n-k)*binomial(n-1,k-1))/k!,k,m,n); /* Vladimir Kruchinin, Sep 27 2012 */

def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number1) # as a square matrix Peter Luschny, Apr 12 2015
# alternatively:

# uses[bell_matrix from A264428]
bell_matrix(lambda n: A029767(n+1), 10) # Peter Luschny, Jan 18 2016

Row sums are A002866.
Second column is A029767.
T(n,m) = n! * Sum_{k=m..n} Stirling1(k,m)*2^(n-k)*binomial(n-1,k-1)/k!, n >= m >= 0. - Vladimir Kruchinin, Sep 27 2012

A360288 Number T(n,k) of permutations of [n] whose excedance set is the k-th finite subset of positive integers in standard order; triangle T(n,k), n>=0, 0<=k<=ceiling(2^(n-1))-1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 3, 7, 1, 3, 1, 1, 1, 15, 7, 31, 3, 17, 7, 15, 1, 7, 3, 7, 1, 3, 1, 1, 1, 31, 15, 115, 7, 69, 31, 115, 3, 37, 17, 69, 7, 37, 15, 31, 1, 15, 7, 31, 3, 17, 7, 15, 1, 7, 3, 7, 1, 3, 1, 1, 1, 63, 31, 391, 15, 245, 115, 675, 7, 145, 69

Offset: 0

Alois P. Heinz, Feb 01 2023

The list of finite subsets of positive integers in standard statistical (or Yates) order begins: {}, {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}, ... cf. A048793, A048794.
The excedance set of permutation p of [n] is the set of indices i with p(i)>i, a subset of [n-1].
All terms are odd.

			T(5,4) = 3: there are 3 permutations of [5] with excedance set {3} (the 4th subset in standard order): 12435, 12534, 12543.
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  3, 1,  1;
  1,  7, 3,  7, 1,  3, 1,  1;
  1, 15, 7, 31, 3, 17, 7, 15, 1, 7, 3, 7, 1, 3, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..15, flattened
Wikipedia, Permutation
Wikipedia, Yates Order

Columns k=0-1 give: A000012, A000225(n-1) for n>=1.
Row sums give A000142.
Row lengths are A011782.
See A152884, A360289 for similar triangles.
Cf. A000027, A029767, A048793, A048794, A263756, A329369.

b:= proc(s, t) option remember; (m->
      `if`(m=0, x^(t/2), add(b(s minus {i}, t+
      `if`(i (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n}, 0)):
seq(T(n), n=0..7);

b[s_, t_] := b[s, t] = Function [m, If[m == 0, x^(t/2), Sum[b[s ~Complement~ {i}, t + If[i < m, 2^i, 0]], {i, s}]]][Length[s]];
T[n_] := CoefficientList[b[Range[n], 0], x];
Table[T[n], {n, 0, 7}]  // Flatten (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

Sum_{k=0..2^(n-1)-1} (k+1) * T(n,k) = A029767(n) for n>=1.

Sum_{k=0..2^(n-1)-1} (2^n-1-k) * T(n,k) = A355258(n+1) for n>=1.

A226968 Number of labeled octopi with n nodes such that each tentacle has an odd number of nodes.

Original entry on oeis.org

0, 1, 1, 8, 30, 264, 1920, 20880, 226800, 3064320, 43908480, 722131200, 12773376000, 249559833600, 5236924492800, 118911189196800, 2883421981440000, 74715282690048000, 2054450584682496000, 59855791774851072000, 1839882143683584000000

Offset: 0

Geoffrey Critzer, Sep 01 2013

nonn

Cf. A029767.

nn=20;f[x_]:=x/(1-x^2);Range[0,nn]!CoefficientList[Series[Log[1/(1-f[x])],{x,0,nn}],x]

For n>=1; (n-1)!*( A000032(n) + (-1)^(n+1) -1 ).

E.g.f.: log(1/(1 - A(x))) where A(x) = x/(1-x^2).

A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

Original entry on oeis.org

1, 3, 1, 14, 9, 1, 90, 83, 18, 1, 744, 870, 275, 30, 1, 7560, 10474, 4275, 685, 45, 1, 91440, 143892, 70924, 14805, 1435, 63, 1, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578

Offset: 1

Vladeta Jovovic, Jan 30 2003

Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling1-triangle A008275(n,k) is unsigned Stirling1-triangle |A008275(n,k)|.
Also the Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
Essentially the same as A131222. - Peter Bala, Feb 12 2022

			Triangle begins
     1;
     3,     1;
    14,     9,    1;
    90,    83,   18,   1;
   744,   870,  275,  30,  1;
  7560, 10474, 4275, 685, 45, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..1225 (rows n = 1..50, flattened).
William Keith, Rishi Nath, and James Sellers, On simultaneous (s, s+t, s+2t, ...)-core partitions, arXiv:2508.00074 [math.CO], 2025. See p. 3.

Cf. A002866 (row sums), A029767 (first column), A131222.

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> n!*(2^(n+1)-1), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, n! (2^(n + 1) - 1)], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

T(n, k) = Sum_{i=k..n} |A008297(n, i)| * |A008275(i, k)|.

E.g.f.: ((1-x)/(1-2*x))^y. - Vladeta Jovovic, Nov 22 2003

A328055 Expansion of e.g.f. -log(1 - x / (1 - x)^2).

Original entry on oeis.org

0, 1, 5, 32, 270, 2904, 38400, 605520, 11113200, 232888320, 5488560000, 143704108800, 4138573824000, 130020673305600, 4425201196416000, 162194862064435200, 6369479157000960000, 266808274486161408000, 11874724379464826880000, 559591797303082672128000

Offset: 0

Ilya Gutkovskiy, Oct 03 2019

a(n) is the number of ways to choose one element from each branch of labeled octopuses with n nodes (cf. A029767 and example below). - Enrique Navarrete, Oct 29 2023

			For n=2, the 3 labeled octopuses are the following, and there are 2+2+1 ways to choose one element from each branch:
O-1-2;
O-2-1;
1-O-2. - _Enrique Navarrete_, Oct 29 2023

Cf. A001906, A004146, A005248, A005443, A029767, A052567 (exponential transform), A100404, A226968, A328054.

[0] cat [Factorial(n - 1)*(Lucas(2*n)-2):n in [1..20]]; // Marius A. Burtea, Oct 03 2019

nmax = 19; CoefficientList[Series[-Log[1 - x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[(n - 1)! (LucasL[2 n] - 2), {n, 1, 19}]]

my(x='x+O('x^20)); concat(0, Vec(serlaplace(-log(1 - x / (1 - x)^2)))) \\ Michel Marcus, Oct 03 2019

E.g.f.: log(1 + Sum_{k>=1} Fibonacci(2*k) * x^k).

a(n) = (n - 1)! * (Lucas(2*n) - 2) for n > 0.

A355257 Array read by ascending antidiagonals. A(n, k) = k! * [x^k] log((1 - x) / (1 - 2*x)) / (1 - x)^n, for 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 5, 14, 0, 1, 7, 29, 90, 0, 1, 9, 50, 206, 744, 0, 1, 11, 77, 406, 1774, 7560, 0, 1, 13, 110, 714, 3804, 18204, 91440, 0, 1, 15, 149, 1154, 7374, 41028, 218868, 1285200, 0, 1, 17, 194, 1750, 13144, 85272, 506064, 3036144, 20603520

Offset: 0

Peter Luschny and Mélika Tebni, Jul 01 2022

Conjecture: For p prime, A(n, p) == -1 (mod p) for n >= 0.
Conjecture: Let n >= 0, k >= 1 and k != 4. Then k divides A(n, k) if and only if k is not prime.
From Mélika Tebni, Jul 04 2022: (Start)
Conjecture: The polynomials of A355259 generate the k+1 column of this array.
Conjecture: For p prime and n even, (A(n, p) / (p - 1)) == 1 (mod p). (End)

			Table A(n, k) begins:
  [0] 0, 1,  3,  14,   90,   744,   7560,    91440,   1285200, ... A029767
  [1] 0, 1,  5,  29,  206,  1774,  18204,   218868,   3036144, ... A103213
  [2] 0, 1,  7,  50,  406,  3804,  41028,   506064,   7084656, ... A355171
  [3] 0, 1,  9,  77,  714,  7374,  85272,  1102968,  15908400, ... A355372
  [4] 0, 1, 11, 110, 1154, 13144, 164136,  2251920,  33923760, ... A355407
  [5] 0, 1, 13, 149, 1750, 21894, 295500,  4320420,  68487120, ... A355414
  [6] 0, 1, 15, 194, 2526, 34524, 502644,  7838928, 131198544, ...
  [7] 0, 1, 17, 245, 3506, 52054, 814968, 13543704, 239548176, ...

Cf. A029767, A103213, A355171, A355259, A355372, A355407, A355414.

egf := n -> log((1 - x)/(1 - 2*x))/(1 - x)^n:
ser := n -> series(egf(n), x, 22):
row := n -> seq(k!*coeff(ser(n), x, k), k = 0..8):
seq(print(row(n)), n = 0..8);
# Alternative:
A := (n, k) -> add(k!*binomial(k + n - 1, k - j - 1)/(j + 1), j = 0..k-1):
seq(print(seq(A(n, k), k = 0..8)), n = 0..7);

A[0, 0] = 0; A[n_, k_] := k! * Binomial[n+k-1, k - 1] * HypergeometricPFQ[{1 - k, 1, 1}, {2, n + 1}, -1];
Table[A[n, k], {n, 0, 8}, {k, 0, 8}] // TableForm

A(n, k) = k!*Sum_{j=0..k-1} binomial(k + n - 1, k - j - 1) / (j + 1).
A(n, k) = k!*Sum_{j=1..k} binomial(n + k - j - 1, n - 1)*(2^j - 1) / j.
A(n, k) = k!*binomial(n + k - 1, k - 1)*hypergeom([1, 1, 1 - k], [2, n + 1], -1) except for A(0, 0) = 0.

cp's OEIS Frontend

A053482 Binomial transform of A029767.

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A052840 a(n) = n*A029767(n-1).

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A053481 First differences of A029767.

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PARI

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A053483 Euler transform of A029767.

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A131222 Exponential Riordan array [1, log((1-x)/(1-2x))].

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A360288 Number T(n,k) of permutations of [n] whose excedance set is the k-th finite subset of positive integers in standard order; triangle T(n,k), n>=0, 0<=k<=ceiling(2^(n-1))-1, read by rows.

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A226968 Number of labeled octopi with n nodes such that each tentacle has an odd number of nodes.

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A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

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