A066524 -id:A066524 - cp's OEIS frontend

A339771 a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^max(i,j).

1, 7, 27, 83, 227, 579, 1411, 3331, 7683, 17411, 38915, 86019, 188419, 409603, 884739, 1900547, 4063235, 8650755, 18350083, 38797315, 81788931, 171966467, 360710147, 754974723, 1577058307, 3288334339, 6845104131, 14227079171, 29527900163, 61203283971

Offset: 0

Bernard Schott, Dec 16 2020

			a(3) = 5*2^4 + 3 = 83.

Eric Billault, Walter Damin, Robert Ferréol, Rodolphe Garin, MPSI Classes Prépas - Khôlles de Maths, Exercices corrigés, Ellipses, 2012, exercice 2.22 (2), pp. 26, 43-44.

Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A142964 (with min instead of max).
Cf. A027981, A066524.
Partial sums of A014480.

Maple
```
seq((2*n-1)*2^(n+1)+3,n=0..40);
```

Table[(2*n - 1)*2^(n + 1) + 3, {n, 0, 29}] (* Amiram Eldar, Dec 16 2020 *)

a(n) = sum(i=0, n, sum(j=0, n, 2^max(i,j))); \\ Michel Marcus, Dec 16 2020

def A339771():
    a, b, c = 1, 7, 27
    yield(a); yield(b)
    while True:
        yield c
        z = 4*a - 8*b + 5*c
        a, b, c = b, c, z
a = A339771()
print([next(a) for  in range(30)]) # _Peter Luschny, Dec 17 2020

a(n) = (2*n-1) * 2^(n+1) + 3.
G.f.: -(2*x+1)/((x-1)*(2*x-1)^2). - Alois P. Heinz, Dec 16 2020
E.g.f: 3*exp(x) + 2*exp(2*x)*(4*x - 1). - Stefano Spezia, Dec 16 2020
a(n) = 2*A066524(n+1) - A142964(n). - Kevin Ryde, Dec 17 2020
a(n) = (2*A027981(n)+1)/3 for n >= 1. - Hugo Pfoertner, Dec 17 2020

A363789 a(n) is the smallest primitive binary Niven number (A363787) whose binary representation is ending with n zeros.

Original entry on oeis.org

1, 6, 60, 2040, 1048560, 137438953440, 1180591620717411303360, 43556142965880123323311949751266331066240, 29642774844752946028434172162224104410437116074403984394101141506025761187823360

Offset: 0

Amiram Eldar, Jun 22 2023

The least term k of A363787 such that A007814(k) = n.
Also, the least binary Niven number (A049445) with a binary weight (A000120) that equals 2^n.
The next term, a(9) = 6.864... * 10^156, is too long to include in the Data section.

Amiram Eldar, Table of n, a(n) for n = 0..11

Subsequence of A049445, A143115 and A363787.

Cf. A000120, A007814, A066524, A358256 (decimal analog).

a[n_] := (2^(2^n)-1) * 2^n; Array[a, 9, 0]

PARI
```
a(n) = (2^(2^n)-1) * 2^n;
```

a(n) = (2^(2^n)-1) * 2^n = A066524(2^n).

a(n) = A143115(2^n).

A135065 A127733 * A007318 as infinite lower triangular matrices.

Original entry on oeis.org

1, 4, 4, 9, 18, 9, 16, 48, 48, 16, 25, 100, 150, 100, 25, 36, 180, 360, 360, 180, 36, 49, 294, 735, 980, 735, 294, 49, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 81, 648, 2268, 4536, 5670, 4536, 2268, 648, 81, 100, 900, 3600, 8400, 12600, 12600, 8400, 3600

Offset: 0

Gary W. Adamson, Nov 16 2007

A135065 * [1/1, 1/2, 1/3, ...] = A066524: (1, 6, 21, 60, 155, ...).

Triangle T(n,k), 0 <= k <= n, read by rows, given by (4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -7/4, 17/28, -32/119, 7/17, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 27 2011

			First few rows of the triangle:
   1;
   4,   4;
   9,  18,   9;
  16,  48,  48,  16;
  25, 100, 150, 100,  25;
  36, 180, 360, 360, 180,  36;
  49, 294, 735, 980, 735, 294,  49;

G. C. Greubel, Table of n, a(n) for the first 50 rows
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

Cf. A000290, A127733, A066524, A014477 (row sums), A084938.

with(combstruct):for n from 0 to 11 do seq(n*m*count(Combination(n), size=m), m = 1 .. n) od; # Zerinvary Lajos, Apr 09 2008

Flatten[Table[Binomial[n,k](n+1)^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jul 12 2013 *)

T(n,k) = binomial(n,k)*(n+1)^2 = A007318(n,k)*A000290(n+1). - Philippe Deléham, Oct 27 2011
T(n-1,k-1) = Sum_{i=-k..k} (-1)^i*(k^2-i^2)*binomial(n,k+i)*binomial(n,k-i). - Mircea Merca, Apr 05 2012
G.f.: (-1 - x - x*y)/(x + x*y - 1)^3. - R. J. Mathar, Aug 12 2015

Corrected by Zerinvary Lajos, Apr 09 2008

A144066 T(n, k) is the number of order-preserving partial transformations (of an n-element chain) of height k (height(alpha) = |Im(alpha)|); triangle T read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 15, 1, 1, 60, 102, 28, 1, 1, 155, 490, 310, 45, 1, 1, 378, 1935, 2220, 735, 66, 1, 1, 889, 6741, 12285, 7315, 1491, 91, 1

Offset: 0

Abdullahi Umar, Sep 09 2008

T(n, k) is also the number of elements in the Green's J-classes of the monoid of order-preserving partial transformations (of an n-element chain). Sum of rows of T(n, k) is A123164.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,   1;
  1,   6,    1;
  1,  21,   15,     1;
  1,  60,  102,    28,    1;
  1, 155,  490,   310,   45,    1;
  1, 378, 1935,  2220,  735,   66,  1;
  1, 889, 6741, 12285, 7315, 1491, 91, 1;
  ...
T(2,1) = 6 because there are exactly 6 order-preserving partial transformations (on a 2-element chain) of height 1, namely: (1)->(1), (1)->(2), (2)->(1), (2)->(2), (1,2)->(1,1), and (1,2)->(2,2) -- the mappings are coordinate-wise.

A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra, 278 (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq., 7 (2004), 04.3.8.

Cf. A000384, A066524, A076454, A112857, A123164.

T(n,k) = C(n,k)*A112857(n,k) for 0 <= k <= n.
C(n-1,k-1)*T(n,k) = 2((n-k+1)/(n-k))*T(n-1,k) + C(n,k)*T(n-1,k-1). [This is wrong.]
From Petros Hadjicostas, Feb 14 2021: (Start)
T(n,k) = 2*n*T(n-1,k)/(n-k) + n*T(n-1,k-1)/k for 1 <= k <= n-1 with T(n,0) = T(n,n) = 1 for n >= 0.
T(n,1) = n*(2^n - 1) = A066524(n) for n >= 1.
T(n,n-1) = n*(2*n - 1) = A000384(n) for n >= 1.
T(n,n-2) = A076454(n-1) for n >= 2. (End)

A372311 Triangle read by rows: T(n, k) = n^k * Sum_{j=0..n} binomial(n - j, n - k) * Eulerian1(n, j).

Original entry on oeis.org

1, 1, 1, 1, 6, 8, 1, 21, 108, 162, 1, 60, 800, 3840, 6144, 1, 155, 4500, 48750, 225000, 375000, 1, 378, 21672, 453600, 4354560, 19595520, 33592320, 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720

Offset: 0

Peter Luschny, Apr 26 2024

			Triangle begins:
  [0] 1;
  [1] 1,   1;
  [2] 1,   6,     8;
  [3] 1,  21,   108,     162;
  [4] 1,  60,   800,    3840,     6144;
  [5] 1, 155,  4500,   48750,   225000,    375000;
  [6] 1, 378, 21672,  453600,  4354560,  19595520,   33592320;
  [7] 1, 889, 94668, 3500658, 60505200, 536479440, 2371803840, 4150656720;

Cf. A061711 (main diagonal), A066524 (column 1), A372312 (row sums).

Cf. A163626, A173018 (eulerian1).

S := (n, k) -> local j; add(eulerian1(n, j)*binomial(n-j, n-k), j = 0..n):
row := n -> local k; seq(S(n, k) * n^k, k = 0..n):
seq(row(n), n = 0..8);

def A372311_row(n) :
    x = polygen(ZZ, 'x')
    A = []
    for m in range(0, n + 1, 1) :
        A.append((-x)^m)
        for j in range(m, 0, -1):
            A[j - 1] = j * (A[j - 1] - A[j])
    return [n^k*c for k, c in enumerate(A[0])]
for n in (0..7) : print(A372311_row(n))

A363849 Triangular array read by rows. T(n,k) is the number of Green's H-classes of rank k in the semigroup of partial transformations, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 18, 1, 1, 60, 150, 40, 1, 1, 155, 900, 650, 75, 1, 1, 378, 4515, 7000, 2100, 126, 1, 1, 889, 20286, 59535, 36750, 5586, 196, 1, 1, 2040, 84700, 435120, 486570, 148176, 12936, 288, 1, 1, 4599, 335880, 2864820, 5358150, 2876202, 493920, 27000, 405, 1

Offset: 0

Geoffrey Critzer, Jun 24 2023

Let H_f denote the H-class in the semigroup of partial transformations containing f. Then H_f contains an idempotent iff the image of f is a transversal for the kernel of f.

Let H_f ~ H_g iff the image of f is contained in the image of g and the kernel of f is more coarse than the kernel of g. Then ~ is a partial order on the H-classes, hence a preorder (quasi-order) on the semigroup. The poset is isomorphic to the Segre product of the Boolean lattice of rank n and the partition lattice of [n+1].

			Triangle begins:
 1;
 1,   1;
 1,   6,   1;
 1,  21,  18,   1;
 1,  60, 150,  40,  1;
 1, 155, 900, 650, 75, 1;
 ...

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, Chapter 4.4 - 4.6.

Wikipedia, Green's relations
Wikipedia, Transversal (combinatorics)

Columns k=0-1 give: A000012, A066524.
Row sums give A134055(n+1).
T(n,n-1) gives A002411.
Cf. A000169, A007318, A008277, A090683, A101818.

T:= (n, k)-> binomial(n, k)*Stirling2(n+1, k+1):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2023

Table[Table[Binomial[n, k] StirlingS2[n + 1, k + 1], {k, 0, n}], {n,0, 5}] // Grid

T(n,k) = A007318(n,k)*A008277(n+1,k+1).
Sum_{k=0..n} T(n,k)*k! = (n+1)^n = A000169(n+1).
T(n,1) = A101818(n,1) = A066524(n) = n*(2^n - 1).  (Every partial function of rank 1 is idempotent.)

cp's OEIS Frontend

A339771 a(n) = Sum_{i=0..n} Sum_{j=0..n} 2^max(i,j).

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A363789 a(n) is the smallest primitive binary Niven number (A363787) whose binary representation is ending with n zeros.

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A135065 A127733 * A007318 as infinite lower triangular matrices.

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A144066 T(n, k) is the number of order-preserving partial transformations (of an n-element chain) of height k (height(alpha) = |Im(alpha)|); triangle T read by rows.

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A372311 Triangle read by rows: T(n, k) = n^k * Sum_{j=0..n} binomial(n - j, n - k) * Eulerian1(n, j).

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A363849 Triangular array read by rows. T(n,k) is the number of Green's H-classes of rank k in the semigroup of partial transformations, n >= 0, 0 <= k <= n.

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