A000629 -id:A000629 - cp's OEIS frontend

A298668 Number T(n,k) of set partitions of [n] into k blocks such that the absolute difference between least elements of consecutive blocks is always > 1; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 3, 0, 1, 7, 2, 0, 1, 15, 12, 0, 1, 31, 50, 6, 0, 1, 63, 180, 60, 0, 1, 127, 602, 390, 24, 0, 1, 255, 1932, 2100, 360, 0, 1, 511, 6050, 10206, 3360, 120, 0, 1, 1023, 18660, 46620, 25200, 2520, 0, 1, 2047, 57002, 204630, 166824, 31920, 720

Offset: 0

Alois P. Heinz, Jan 24 2018

			T(5,1) = 1: 12345.
T(5,2) = 7: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345.
T(5,3) = 2: 124|3|5, 12|34|5.
T(7,4) = 6: 1246|3|5|7, 124|36|5|7, 124|3|56|7, 126|34|5|7, 12|346|5|7, 12|34|56|7.
T(9,5) = 24: 12468|3|5|7|9, 1246|38|5|7|9, 1246|3|58|7|9, 1246|3|5|78|9, 1248|36|5|7|9, 124|368|5|7|9, 124|36|58|7|9, 124|36|5|78|9, 1248|3|56|7|9, 124|38|56|7|9, 124|3|568|7|9, 124|3|56|78|9, 1268|34|5|7|9, 126|348|5|7|9, 126|34|58|7|9, 126|34|5|78|9, 128|346|5|7|9, 12|3468|5|7|9, 12|346|58|7|9, 12|346|5|78|9, 128|34|56|7|9, 12|348|56|7|9, 12|34|568|7|9, 12|34|56|78|9.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1,    1;
  0, 1,    3;
  0, 1,    7,     2;
  0, 1,   15,    12;
  0, 1,   31,    50,     6;
  0, 1,   63,   180,    60;
  0, 1,  127,   602,   390,    24;
  0, 1,  255,  1932,  2100,   360;
  0, 1,  511,  6050, 10206,  3360,  120;
  0, 1, 1023, 18660, 46620, 25200, 2520;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wenqin Cao, Emma Yu Jin, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy

Columns k=0-11 give (offsets may differ): A000007, A057427, A168604, A028243, A028244, A028245, A032180, A228909, A228910, A228911, A228912, A228913.
Row sums give A229046(n-1) for n>0.
T(2n+1,n+1) gives A000142.
T(2n,n) gives A001710(n+1).
Cf. A000629, A000670, A008277, A028246, A048993, A076726, A110654, A136011, A173018.

b:= proc(n, m, t) option remember; `if`(n=0, x^m, add(
      b(n-1, max(m, j), `if`(j>m, 1, 0)), j=1..m+1-t))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);
# second Maple program:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), (k-1)!*Stirling2(n-k+1, k)):
seq(seq(T(n, k), k=0..ceil(n/2)), n=0..14);
# third Maple program:
T:= proc(n, k) option remember; `if`(k<2, `if`(n=0 xor k=0, 0, 1),
      `if`(k>ceil(n/2), 0, add((k-j)*T(n-1-j, k-j), j=0..1)))
    end:
seq(seq(T(n, k), k=0..ceil(n/2)), n=0..14);

T[n_, k_] := T[n, k] = If[k < 2, If[Xor[n == 0, k == 0], 0, 1],
     If[k > Ceiling[n/2], 0, Sum[(k-j) T[n-1-j, k-j], {j, 0, 1}]]];
Table[Table[T[n, k], {k, 0, Ceiling[n/2]}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 08 2021, after third Maple program *)

T(n,k) = (k-1)! * Stirling2(n-k+1,k) for k>0, T(n,0) = A000007(n).
T(n,k) = Sum_{j=0..k-1} (-1)^j*C(k-1,j)*(k-j)^(n-k) for k>0, T(n,0) = A000007(n).
T(n,k) = (k-1)! * A136011(n,k) for n, k >= 1.
Sum_{j>=0} T(n+j,j) = A076726(n) = 2*A000670(n) = A000629(n) + A000007(n).

A299907 Number of decomposable lonesum n X n (0,1) matrices.

Original entry on oeis.org

1, 2, 16, 344, 13528, 833432, 73871416, 8893109864, 1394602938808, 275985896665592, 67227147723919096, 19756312865302785224, 6889612105281125091448, 2811911251972519547757272, 1327454489179890318710048056, 717555570945004782603934710824

Offset: 0

N. J. A. Sloane, Feb 23 2018

nonn

Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017. Also Discrete Math., 341 (2018), 341-349.

Main diagonal of A299906.
See also A000629, A221961 for symmetric square lonesum matrices.
See A099594 for lonesum (0,1) matrices.

a[n_] := Sum[Binomial[j - 1 , k - 1 ]*j!^2*StirlingS2[n + 1, j + 1]^2/k!, {k, 0, n}, {j, k, n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Feb 24 2018 *)

More terms from Jean-François Alcover, Feb 24 2018

A300693 a(n) = number of edges in a concertina n-cube.

Original entry on oeis.org

0, 1, 6, 42, 344, 3230, 34452

Offset: 0

Tilman Piesk, Apr 03 2018

n-place formulas in first-order logic like Ax Ey P(x, y) can be ordered by implication. This Hasse diagram has A000629(n) vertices and a(n) edges.
This is the second diagonal on the right in A300700, the triangle of faces in the concertina n-cube.
The corresponding sequence for cocoon concertina n-cubes, which have more internal vertices and edges, is A300694.

Tilman Piesk, Formulas in predicate logic (Wikiversity)
Tilman Piesk, Image of a concertina square with 6 and cube with 42 edges
Tilman Piesk, Lists of edges for n=2..5
Tilman Piesk, Python code used to generate the sequence

Cf. A300700, A000629, A300694.

a(n) = A300700(n, n-1).

A300699 Irregular triangle read by rows: T(n, k) = number of vertices with rank k in concertina n-cube.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 6, 6, 3, 1, 1, 4, 12, 18, 28, 24, 28, 18, 12, 4, 1, 1, 5, 20, 40, 80, 95, 150, 150, 150, 150, 95, 80, 40, 20, 5, 1, 1, 6, 30, 75, 180, 270, 506, 660, 840, 1080, 1035, 1035, 1080, 840, 660, 506, 270, 180, 75, 30, 6, 1, 1, 7, 42, 126, 350, 630, 1337, 2107, 3192, 4760

Offset: 0

Tilman Piesk, Mar 11 2018

n-place formulas in first-order logic like Ax Ey P(x, y) ordered by implication form a graded poset, and its Hasse diagram is the concertina n-cube.
Sum of row n is A000629(n), the number of vertices of a concertina n-cube.
The rows are palindromic. Their lengths are the central polygonal numbers A000124 = 1, 2, 4, 7, 11, 16, 22, ... That means after row 0 rows of even and odd length follow each other in pairs.
The central values are 1, (1), (2), 6, 24, (150), (1035), 9030, 88760, (1002204), ... (Values next to the center in rows of even length are in parentheses.)
Maximal values are 1, 1, 2, 6, 28, 150, 1080, 9030, 88760, 1002204, ...
A300695 is a triangle of the same shape that shows the number of ranks in cocoon concertina hypercubes.

			First rows of the triangle:
    k   0   1   2   3   4   5    6    7    8    9   10  11  12  13  14  15
  n
  0     1
  1     1   1
  2     1   2   2   1
  3     1   3   6   6   6   3    1
  4     1   4  12  18  28  24   28   18   12    4    1
  5     1   5  20  40  80  95  150  150  150  150   95  80  40  20   5   1
  6     1   6  30  75 180 270  506  660  840 1080 1035 ...
The ranks of vertices of a concertina cube (n=3) can be seen in the linked Hasse diagrams. T(3, 4) = 6, so there are 6 vertices with rank 4.
Ey Ez Ax P(x, y, z) implies Ey Ax Ez P(x, y, z), and their ranks are 3 and 4. As the difference in rank is 1, this implication is an edge in the Hasse diagram.

Tilman Piesk, Rows 0..9, flattened
Tilman Piesk, Rows 0..9
Tilman Piesk, Formulas in predicate logic (Wikiversity)
Tilman Piesk, Concertina cube Hasse diagram with labels and with highlighted ranks
Tilman Piesk, Lists of vertices ordered by rank for n=2..6
Tilman Piesk, Python code used to generate the sequence

Cf. A000124, A000629, A300695.

A300700 Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 18, 42, 26, 1, 58, 252, 344, 150, 1, 190, 1420, 3380, 3230, 1082, 1, 614, 7770, 29200, 47130, 34452, 9366

Offset: 0

Tilman Piesk, Mar 11 2018

n-place formulas in first-order logic like Ax Ey P(x, y) can be ordered by implication. This Hasse diagram can be interpreted as an n-dimensional convex polytope with face dimensions ranging from 0 (the vertices) to n (the polytope itself).
The right diagonal (n-k = 0, number of vertices) is A000629, which is twice an ordered Bell number (A000670) for n>0.
The second right diagonal (n-k = 1, number of edges) is A300693.
The second left diagonal (k = 1, number of facets) is 2, 6, 18, 58, 190, 614, ... (not to be confused with A151282 or A193777).
The third left diagonal (k = 2, number of ridges) is 6, 42, 252, 1420, 7770, ...
The row sums are A300701. The central diagonal starts 1, 6, 252, 29200 and the row maxima start 1, 2, 6, 42, 344, 3380, 47130.
The corresponding triangle for hypercubes is A013609, and its row sums are A000244 (powers of 3). That for permutohedra is A019538, and its row sums are A000670 (ordered Bell numbers).

			First rows of the triangle:
  k      0     1     2     3     4     5     6         sums = A300701
n
0        1                                                1
1        1     2                                          3
2        1     6     6                                   13
3        1    18    42    26                             87
4        1    58   252   344   150                      805
5        1   190  1420  3380  3230  1082               9303
6        1   614  7770 29200 47130 34452  9366       128533
T(3, 3-1) = T(3, 2) = 42 is the number of 1-faces (edges) of the concertina 3-cube. It has 26 vertices, 42 edges, 18 faces and 1 cell.
In the reflected triangle the column number is the dimension of the counted faces:
  n-k    0     1     2     3     4     5     6
n
0        1
1        2     1
2        6     6     1
3       26    42    18     1
4      150   344   252    58     1
5     1082  3230  3380  1420   190     1
6     9366 34452 47130 29200  7770   614     1

Tilman Piesk, Formulas in predicate logic (Wikiversity)
Tilman Piesk, Skeleton and solid representation of a concertina cube
Tilman Piesk, SAGE code used to generate the sequence

Cf. A300701, A000629, A300693.
Cf. A013609, A000244 (for hypercubes).
Cf. A019538, A000670 (for permutohedra).

A363732 Triangle read by rows. The triangle algorithm applied to (-1)^n/n!.

Original entry on oeis.org

1, -2, 1, 5, -4, 1, -15, 15, -6, 1, 52, -60, 30, -8, 1, -203, 260, -150, 50, -10, 1, 877, -1218, 780, -300, 75, -12, 1, -4140, 6139, -4263, 1820, -525, 105, -14, 1, 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1, -115975, 190323, -149040, 73668, -25578, 6552, -1260, 180, -18, 1

Offset: 0

Peter Luschny, Jun 18 2023

The triangle algorithm, as understood here, is a transformation that maps a sequence of integers (a(n) : n >= 0) to a polynomial sequence. A polynomial sequence is a sequence of polynomials (P(n,x) : n >= 0) with degree(P(n, x)) = n for all n >= 0.
The polynomials P(n, x) are recursively defined by P(n, x) = p(n, 0, x), where the initial sequence is p(0, m, x) = a(m), and for n > 0 is given by
  p(n, m, x) = (m + 1)*p(n - 1, m + 1, x) - (m + 1 - x)*p(n - 1, m, x).
Here we identify the polynomial sequence with the infinite lower triangular array of its coefficients, T(n, k) = [x^k] P(n, x). We call the mapping (a(n) : n >= 0) -> (T(n, k) : 0 <= k <= n) the 'triangle algorithm', following the lead of Kawasaki and Ohno.
Evaluating P(n, x) at different values of x gives rise to a multitude of other sequences; in particular, the transformation a(n) -> b(n) = P(n, 1) will be called the Akiyama-Tanigawa transform of a.
The triangle algorithm was studied by Akiyama and Tanigawa, Chen, Imatomi, Arakawa and Kaneko, Kawasaki and Ohno, and others, at first in connection with the Bernoulli and Poly-Bernoulli numbers.
.
The paradigmatic examples are:
  a(n) = 1 -> x^n, the standard base of polynomials, A023531.
  a(n) = n + 1 -> binomial(n, k), Pascal triangle, A007318.
  a(n) = n + 1 -> P(n, 1) powers of 2, A000079.
  a(n) = n + 1 -> P(n, 0) the all 1's sequence A000012.
  a(n) = 2^n -> [x^k] P(n, x), A154921.
  a(n) = 2^n -> P(n, 0) Fubini numbers, A000670.
  a(n) = 2^n -> P(n, 1) ordered set partitions of subsets of [n], A000629.
  a(n) = 2^n -> P(n,-1) osp. of [n] with even number of blocks, A052841.
  a(n) = 1 / (n + 1) -> [x^k] B(n, x), Bernoulli polynomials, A196838/A196839.
  a(n) = 1 / (n + 1) -> B(n, 1), the Bernoulli numbers, A164555/A027642.
  a(n) = Chen(n) -> skp(n, x), Swiss-Knife polynomials, A153641.
  a(n) = Chen(n) -> P(n, 0), 2^n*Euler(n, 1/2) = Euler(n), A122045.
  a(n) = Chen(n) -> P(n, 1), 2^n*Euler(n, 1), A155585.
  a(n) = (-1)^n/n! -> [x^k] P(n, x) this "Bell" triangle.
  a(n) = (-1)^n/n! -> (-1)^n*P(n, 1) = Bell(n), A000110.
  a(n) = (-1)^n/n! -> (-1)^n*P(n,-1) = 2-Bell(n), A005493.
  a(n) = 1/n! -> (-1)^n*P(n, 1) = complementary Bell(n), A000587.
  a(n) = 1/n! -> (-1)^n*P(n,-1) = complementary 2-Bell(n), A074051.
  (For Chen's sequence see A363524.)
.
The present sequence deals with the case of the Bell numbers. In contrast to Aitken's array A011971 and its variants A123346 and A011972, the Bell numbers do not appear as a column of the triangle but as row sums (times (-1)^n), i.e., as values of the associated polynomials at x = 1. Comparing this with a similar situation with the Bernoulli numbers/polynomials, our triangle could be viewed as a more organic generalization of the Bell numbers. Indeed, the names 'Bell triangle' and 'Bell polynomials' would be justified here; but these are already assigned to other concepts.

			The triangle T(n, k) starts:
  [0]     1;
  [1]    -2,      1;
  [2]     5,     -4,     1;
  [3]   -15,     15,    -6,      1;
  [4]    52,    -60,    30,     -8,    1;
  [5]  -203,    260,  -150,     50,  -10,    1;
  [6]   877,  -1218,   780,   -300,   75,  -12,   1;
  [7] -4140,   6139, -4263,   1820, -525,  105, -14,   1;
  [8] 21147, -33120, 24556, -11368, 3640, -840, 140, -16, 1;

Peter Luschny, Table of n, a(n) for row 0..100.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Naho Kawasaki and Yasuo Ohno, The triangle algorithm for Bernoulli polynomials, Integers, vol. 23 (2023). (See figure 4.)
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A293037 (row sums), A000110 (row sums, unsigned), A005493 (alternating row sums, signed).

Cf. A023531, A007318, A000079, A000012, A154921, A000670, A196838/A196839, A164555/A027642, A153641, A122045, A155585, A011971, A123346, A011972, A056857, A363524 (Chen sequence).

TA := proc(a, n, m, x) option remember; if n = 0 then a(m) else
normal((m + 1)*TA(a, n - 1, m + 1, x) - (m + 1 - x)*TA(a, n - 1, m, x)) fi end:
seq(seq(coeff(TA(n -> (-1)^n/n!, n, 0, x), x, k), k = 0..n), n = 0..10);

(* rows[0..n], n[0..oo] *)
(* row[n]= *)
n=9;r={};For[a=n+1,a>0,a--,AppendTo[r,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j))/(2*j+2),{j,0,n-a}]]];r
(* columns[1..n], n[0..oo] *)
(* column[n]= *)
n=0;c={};For[a=1,a<15,a++,AppendTo[c,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j-1))/(2*j),{j,1,n}]]];c
(* sequence *)
s={};For[n=0,n<15,n++,For[a=n+1,a>0,a--,AppendTo[s,(-1)^(a+1)*Sum[StirlingS2[a,k],{k,0,a}]*Product[(2*(a+j))/(2*j+2),{j,0,n-a}]]]];s
(* Detlef Meya, Jun 22 2023 *)

def a(n): return (-1)^n / factorial(n)
@cached_function
def p(n, m):
    R = PolynomialRing(QQ, "x")
    if n == 0: return R(a(m))
    return R((m + 1)*p(n - 1, m + 1) - (m + 1 - x)*p(n - 1, m))
for n in range(10): print(p(n, 0).list())

A054255 Triangle T(n,k) (n >= 1, 0<=k<=n) giving number of preferential arrangements of n things beginning with k (transposed, then read by rows).

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 6, 18, 25, 26, 24, 84, 134, 149, 150, 120, 480, 870, 1050, 1081, 1082, 720, 3240, 6600, 8700, 9302, 9365, 9366, 5040, 25200, 57120, 82320, 92526, 94458, 94585, 94586, 40320, 221760, 554400, 871920, 1038744, 1085364, 1091414, 1091669, 1091670

Offset: 1

Eugene McDonnell (Eemcd(AT)aol.com), May 05 2000

Can be generated from Stirling_2 triangle A008277 (cf. A028246, which is intermediate between the two arrays).

			   1;
   1,  2;
   2,  5,   6;
   6, 18,  25,  26;
  24, 84, 134, 149, 150;
  ...

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Row sums give A000670. First 3 rows are A000629, A002050 = A000629 - 1, 2*A002051 = (A000629 - 2^m) (m >= 0).

Cf. A090665 (triangle with rows reversed).

More terms from James Sellers, May 05 2000

A123114 a(n) = Sum_{r>0,s>0} binomial(r*s-1,n-1)/2^(r+s).

Original entry on oeis.org

1, 3, 13, 83, 701, 7363, 92541, 1354627, 22636861, 425241347, 8871085565, 203487078403, 5090418231549, 137920771272963, 4023549748488445, 125743894742698243, 4191213031967650813, 148414827031140706307

Offset: 1

Vladeta Jovovic, Sep 28 2006

Vaclav Kotesovec, Table of n, a(n) for n = 1..400
M. Maia and M. Mendez, On the arithmetic product of combinatorial species

Cf. A101370, A000629.

Table[Sum[StirlingS1[n, k]*(Sum[(j - 1)!*StirlingS2[k, j], {j, 1, k}])^2, {k, 1, n}]/(n-1)!, {n, 1, 20}] (* Vaclav Kotesovec, Jun 07 2019 *)
Table[-(-1)^n + Sum[StirlingS1[n, k]*PolyLog[1-k, 2]^2, {k, 2, n}]/(n-1)!, {n, 1, 20}] (* Vaclav Kotesovec, Jun 07 2019 *)

a(n) = (1/(n-1)!)*Sum_{k=1..n} Stirling1(n,k)*b(k)^2, where b(n) = Sum_{k=1..n} (k-1)!*Stirling2(n,k).

a(n) ~ c * (n-1)! / (log(2))^(2*n), where c = 2^(-log(2)/2) = 0.7864497045594053649114085152934509198700275589579678941719548714254307448... - Vaclav Kotesovec, Jun 07 2019

A129064 Fourth column (m=3) of triangle A129062 and third column of triangle A079641.

Original entry on oeis.org

1, 12, 120, 1230, 13650, 166376, 2229444, 32724810, 523531470, 9080409492, 169892449584, 3412891866566, 73300097535210, 1676670468061920, 40704197313912060, 1045464783485987298, 28328001168991093350

Offset: 0

Wolfdieter Lang, May 04 2007

A129063, A000629 give m=2, m=1 columns.

Cf. A079641, A129062.

a(n) = A129062(n+3,3), n>=0.
a(n) = A079641(n+2,2), n>=0.
E.g.f.: (d^3/dx^3)*((-log(2-exp(x)))^3)/3!.

A176785 Sequence with e.g.f. g(x) = -(1/2)sqrt(2exp(-2*x)-1) + 1/2.

Original entry on oeis.org

0, 1, 0, 4, 24, 256, 3360, 53824, 1016064, 22095616, 543966720, 14955833344, 454227400704, 15103031627776, 545668238868480, 21286707282264064, 891735287528914944, 39926103010743156736

Offset: 0

Karol A. Penson, Apr 26 2010

			a(4) = 24: The 24 plane increasing trees on 4 vertices are
............................................................
.........1(x4 colors).......1(x4 colors).......1(x4 colors).
......../|\................/|\................/|\...........
......./.|.\............../.|.\............../.|.\..........
......2..3..4............2..4..3............3..2..4.........
............................................................
.........1(x4 colors).......1(x4 colors).......1(x4 colors).
......../|\................/|\................/|\...........
......./.|.\............../.|.\............../.|.\..........
......3..4..2............4..2..3............4..3..2.........
............................................................

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.

Cf. A000629, A000984, A124212, A124214, A229558.

max = 17; g[x_] := -(1/2)*Sqrt[2*Exp[-2*x] - 1] + 1/2; CoefficientList[ Series[ g[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Oct 05 2011 *)

x='x+O('x^66); concat ([0], Vec( serlaplace( serreverse( -1/2*log(1-2*x+2*x^2) ) ) ) ) \\ Joerg Arndt, Mar 01 2014

The e.g.f. A(x) satisfies the autonomous differential equation
A' = (1-2*A+2*A^2)/(1-2*A) with A(0) = 0. The compositional inverse of the e.g.f. is -1/2*log(1-2*x+2*x^2).
a(n) = (-1)^(n-1)*D^(n-1)(1) evaluated at x = 1, where D denotes the operator g(x) -> d/dx((x+1/x)*g(x)).
Applying [Bergeron et al., Theorem 1] to the result x = int {t = 0..A(x)} 1/phi(t), where phi(t) = (1-2*t+2*t^2)/(1-2*t) = 1+2*t^2+4*t^3+8*t^4+... leads to the following combinatorial interpretation for this sequence: a(n) gives the number of plane increasing trees on n vertices with no vertices of outdegree 1 and where each vertex of outdegree k >= 2 can be colored in 2^(k-1) ways. An example is given below. - Peter Bala, Sep 06 2011
a(n) ~ 2^(n-3/2)*n^(n-1)/(exp(n)*(log(2))^(n-1/2)). - Vaclav Kotesovec, Jun 28 2013
a(n+1) = 1/sqrt(2) * Sum_{k >= 0} (1/8)^k*binomial(2*k,k)*(2*k - 1)^n = 1/sqrt(2)*Sum_{k >= 0} (-1/2)^k*binomial(-1/2,k)*(2*k - 1)^n = Sum_{k = 0..n} Sum_{i = 0..k} (-1)^(k-i)/4^k* binomial(2*k,k)*binomial(k,i)*(2*i - 1)^n. Cf. A124212, A124214 and A229558. - Peter Bala, Aug 30 2016

cp's OEIS Frontend

A298668 Number T(n,k) of set partitions of [n] into k blocks such that the absolute difference between least elements of consecutive blocks is always > 1; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A299907 Number of decomposable lonesum n X n (0,1) matrices.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A300693 a(n) = number of edges in a concertina n-cube.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A300699 Irregular triangle read by rows: T(n, k) = number of vertices with rank k in concertina n-cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A300700 Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A363732 Triangle read by rows. The triangle algorithm applied to (-1)^n/n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

A054255 Triangle T(n,k) (n >= 1, 0<=k<=n) giving number of preferential arrangements of n things beginning with k (transposed, then read by rows).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A123114 a(n) = Sum_{r>0,s>0} binomial(r*s-1,n-1)/2^(r+s).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A129064 Fourth column (m=3) of triangle A129062 and third column of triangle A079641.

Views

Author

Keywords

Crossrefs

A176785 Sequence with e.g.f. g(x) = -(1/2)sqrt(2exp(-2*x)-1) + 1/2.