A045623 -id:A045623 - cp's OEIS frontend

A160232 Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n.

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 8, 1, 5, 14, 25, 28, 16, 1, 6, 20, 44, 66, 64, 32, 1, 7, 27, 70, 129, 168, 144, 64, 1, 8, 35, 104, 225, 360, 416, 320, 128, 1, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 1, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512, 1, 11, 65

Offset: 1

N. J. A. Sloane, May 15 2010

Suggested by a question from Phyllis Chinn (Humboldt State University).
As triangle, mirror image of A105306. - Philippe Deléham, Nov 01 2011
A160232 is jointly generated with A208341 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + 2x*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Feb 25 2012
Subtriangle of the triangle T(n,k) given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012

			Array begins:
  1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, ...
  1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, ...
  1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, 13056, 29952, 68096, 153600, 344064, 765952, 1695744, 3735552, ...
  1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, ...
  1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, 236640, 632448, 1661056, 4296192, 10961664, 27630592, ...
From _Clark Kimberling_, Feb 25 2012: (Start)
As a triangle (see Comments):
  1;
  1,  1;
  1,  2,  2;
  1,  3,  5,  4;
  1,  4,  9, 12,  8;  (End)
From _Philippe Deléham_, Mar 08 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  2,  0;
  1,  3,  5,  4,  0;
  1,  4,  9, 12,  8,  0;
  1,  5, 14, 25, 28, 16,  0; (End)

Rows give A011782, A045623, A058396, A062109, A169792-A169797.

Cf. A062110, A105306, A208341.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2*x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A160232 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208341 *)
(* Clark Kimberling, Feb 25 2012 *)

From Philippe Deléham, Mar 08 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-2*y*x)/(1-2*y*x-x+y*x^2).
Sum_{k=0..n, n>0} T(n,k)*x^k = A000012(n), A001519(n), A052984(n-1) for x = 0, 1, 2 respectively. (End)

A169792 Expansion of ((1-x)/(1-2x))^5.

Original entry on oeis.org

1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, 23336960, 55623680, 131563520, 309002240, 721092608, 1672806400, 3859415040, 8859156480, 20240138240, 46038777856, 104291368960, 235342397440, 529153392640

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 4 parts equal to 0. - Milan Janjic, Jun 27 2010

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^5; see A291000. - Clark Kimberling, Aug 24 2017

Muniru A Asiru, Table of n, a(n) for n = 0..500
Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).

((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

Concatenation([1],List([1..30],n->2^n*(n+4)*(n^3+26*n^2+171*n+186)/768)); # Muniru A Asiru, Aug 22 2018

seq(coeff(series(((1-x)/(1-2*x))^5, x,n+1),x,n),n=0..30); # Muniru A Asiru, Aug 22 2018

CoefficientList[Series[((1 - x)/(1 - 2 x))^5, {x, 0, 28}], x] (* Michael De Vlieger, Oct 15 2018 *)

Vec(((1-x)/(1-2*x))^5+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5), n >= 6. - Vincenzo Librandi, Mar 14 2011

a(n) = 2^n*(n+4)*(n^3 + 26*n^2 + 171*n + 186)/768, n > 0. - R. J. Mathar, Mar 14 2011

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 5, 4, 1, 4, 12, 13, 6, 1, 8, 28, 38, 25, 8, 1, 16, 64, 104, 88, 41, 10, 1, 32, 144, 272, 280, 170, 61, 12, 1, 64, 320, 688, 832, 620, 292, 85, 14, 1, 128, 704, 1696, 2352, 2072, 1204, 462, 113, 16, 1

Offset: 0

Philippe Deléham, Dec 05 2011

Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).

			Triangle begins:
  1;
  0,  1;
  1,  2,  1;
  2,  5,  4,  1;
  4, 12, 13,  6,  1;
  8, 28, 38, 25,  8,  1;

Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.

Diagonals: A000012, A005843, A001844, A035597,
Columns: A034008, A045623, A084851, A055585,
Row sums: A052156

CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
G.f.: (1-x)^2/(1-(y+2)*x).

A169797 Expansion of ((1-x)/(1-2x))^10.

Original entry on oeis.org

1, 10, 65, 340, 1550, 6412, 24650, 89440, 309605, 1030490, 3317445, 10377180, 31655820, 94451520, 276313200, 794169792, 2246410560, 6262748160, 17230138880, 46831339520, 125870737408, 334826700800, 882159984640, 2303540756480, 5965195018240, 15327324667904

Offset: 0

N. J. A. Sloane, May 15 2010

a(n) is the number of weak compositions of n with exactly 9 parts equal to 0. - Milan Janjic, Jun 27 2010

Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (20,-180,960,-3360,8064,-13440,15360,-11520,5120,-1024)

((1-x)/(1-2x))^k: A011782, A045623, A058396, A062109, A169792-A169797; a row of A160232.

CoefficientList[Series[((1-x)/(1-2x))^10,{x,0,30}],x] (* or *) Join[ {1}, LinearRecurrence[{20,-180,960,-3360,8064,-13440,15360,-11520,5120,-1024},{10,65,340,1550,6412,24650,89440,309605,1030490,3317445},30]] (* Harvey P. Dale, Aug 21 2014 *)

Vec(((1-x)/(1-2*x))^10+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = 2^(n-17)*(n+11) *(n^8 + 124*n^7 + 5986*n^6 + 143944*n^5 + 1836529*n^4 + 12358156*n^3 + 42005484*n^2 + 64730736*n + 33747840)/2835, n > 0. - R. J. Mathar, Mar 14 2011

A221876 T(n,k) is the number of order-preserving full contraction mappings (of an n-chain) with exactly k fixed points.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 12, 5, 2, 1, 28, 12, 5, 2, 1, 64, 28, 12, 5, 2, 1, 144, 64, 28, 12, 5, 2, 1, 320, 144, 64, 28, 12, 5, 2, 1, 704, 320, 144, 64, 28, 12, 5, 2, 1, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1

Offset: 1

Abdullahi Umar, Feb 28 2013

Row sum is A001792(n-1).
The matrix inverse starts
1;
-2,1;
-1,-2,1;
0,-1,-2,1;
1,0,-1,-2,1;
2,1,0,-1,-2,1;
3,2,1,0,-1,-2,1;
4,3,2,1,0,-1,-2,1;
5,4,3,2,1,0,-1,-2,1;
6,5,4,3,2,1,0,-1,-2,1;
7,6,5,4,3,2,1,0,-1,-2,1; - R. J. Mathar, Apr 12 2013
...
T(n,k) is also the total number of occurrences of parts k in all compositions (ordered partitions) of n, see example. The equivalent sequence for partitions is A066633. Omar E. Pol, Aug 26 2013

			T(5,3) = 5 because there are exactly 5 order-preserving full contraction mappings (of a 5-chain) with exactly 3 fixed points, namely: (12333), (12334), (22344), (23345), (33345).
Triangle begins:
1,
2, 1,
5, 2, 1,
12, 5, 2, 1,
28, 12, 5, 2, 1,
64, 28, 12, 5, 2, 1,
144, 64, 28, 12, 5, 2, 1,
320, 144, 64, 28, 12, 5, 2, 1,
704, 320, 144, 64, 28, 12, 5, 2, 1,
1536, 704, 320, 144, 64, 28, 12, 5, 2, 1,
3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1,
...
Note that column k is column 1 shifted down by k positions.
Row 4 is [12, 5, 2, 1]: in the compositions of 4
[ 1]  [ 1 1 1 1 ]
[ 2]  [ 1 1 2 ]
[ 3]  [ 1 2 1 ]
[ 4]  [ 1 3 ]
[ 5]  [ 2 1 1 ]
[ 6]  [ 2 2 ]
[ 7]  [ 3 1 ]
[ 8]  [ 4 ]
there are 12 parts=1, 5 parts=2, 2 part=3, and 1 part=4.
- _Joerg Arndt_, Sep 01 2013

A. D. Adeshola, V. Maltcev and A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain, arXiv:1303.7428 [math.CO], 2013.
A. D. Adeshola, A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain, JMCC 106 (2017) 37-49

Cf. A001792, A221877, A221878, A221879, A221880, A221881, A221882.

T[n_, n_] = 1; T[n_, k_] := (n - k + 3)*2^(n - k - 2);
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 21 2018 *)

T(n,n) = 1, T(n,k) = (n-k+3)*2^(n-k-2) for n>=2 and n > k > 0.
T(2*n+1,n+1) = T(n+1,1) = A045623(n) for n>=0.
T(n,k) = A045623(n-k), n>=1, 1<=k<=n. - Omar E. Pol, Sep 01 2013

A079859 a(n) = n*2^(n-4).

Original entry on oeis.org

4, 10, 24, 56, 128, 288, 640, 1408, 3072, 6656, 14336, 30720, 65536, 139264, 294912, 622592, 1310720, 2752512, 5767168, 12058624, 25165824, 52428800, 109051904, 226492416, 469762048, 973078528, 2013265920, 4160749568, 8589934592, 17716740096, 36507222016

Offset: 4

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

a(n) = the number of occurrences of 3s in the palindromic compositions of m = 2*n-1 = the number of occurrences of 4s in the palindromic compositions of k = 2*n.
This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862, A079863. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.
Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004
a(n) appears to be the coefficient of Pi^n in the closed-form expression for the expected value of X^n, where X is the area of a spherical triangle formed by three random points on the unit sphere. (The n*2^(n-4) formula applies when n=2,3 as well, and produces fractional coefficients.) - Drake Thomas, Jan 24 2021

			a(4)=4 since the palindromic compositions of 7 that contain a 3 are 2+3+2, 1+1+3+1+1 and 3+1+3, for a total of 4 3s. The palindromic compositions of 8 that contain a 4 are 2+4+2, 1+1+4+1+1 and 4+4.

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Math StackExchange, The distribution of areas of a random triangle on the sphere.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A078836, A079861, A079862.

Main diagonal of A049089.

[n*2^(n-4) : n in [4..40]]; // Vincenzo Librandi, Sep 22 2011

Mathematica
```
Table[i*2^(i - 4), {i, 4, 50}]
```

Vec(-2*x^4*(3*x-2)/(2*x-1)^2 + O(x^50)) \\ Colin Barker, Sep 29 2015

a(n) = n*2^(n-4);
vector(40, n, a(n+3)) \\ Altug Alkan, Sep 29 2015

O.g.f.: 2*x^4*(2-3*x)/(1-2*x)^2. a(n) = 2*A045623(n-3). - R. J. Mathar, Jun 13 2008
a(n) = 4*a(n-1) - 4*a(n-2) for n>5. - Colin Barker, Sep 29 2015
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=4} 1/a(n) = 16*log(2) - 32/3.
Sum_{n>=4} (-1)^n/a(n) = 20/3 - 16*log(3/2). (End)
E.g.f.: x*(exp(2*x) - 1 - 2*x - 2*x^2)/8. - Stefano Spezia, Apr 06 2021

A108244 Triangle read by rows: row n gives list of all compositions of n ordered first by decreasing length, then by reverse colexicographical order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 2, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 1, 1, 4, 2, 3, 3, 2, 4, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 1

Hugo van der Sanden, Jun 20 2005

An example of a sequence which contains all finite sequences of positive integers as subsequences.
From Andrey Zabolotskiy, May 18 2018: (Start)
At first, the ordering within the compositions of fixed length coincides with the lexicographical order (which is the case of A228369), but for n = 5 the partitions {2, 1, 2}, {1, 3, 1}, {2, 2, 1} go in this order because the order becomes reverse lexicographical when they are reversed (read right-to-left): {2, 1, 2}, {1, 3, 1}, {1, 2, 2}.
Length of k-th composition is A124748(k-1)+1.
Reversing every composition gives A296772. (End)

			The first 5 rows are:
{1}
{1, 1}, {2}
{1, 1, 1}, {1, 2}, {2, 1}, {3}
{1, 1, 1, 1}, {1, 1, 2}, {1, 2, 1}, {2, 1, 1}, {1, 3}, {2, 2}, {3, 1}, {4}
{1, 1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 2, 1}, {1, 2, 1, 1}, {2, 1, 1, 1}, {1, 1, 3}, {1, 2, 2}, {2, 1, 2}, {1, 3, 1}, {2, 2, 1}, {3, 1, 1}, {1, 4}, {2, 3}, {3, 2}, {4, 1}, {5}

Eric Weisstein's World of Mathematics, Combinatorial composition

Cf. A045623, A124748.

Triangles of compositions: A066099 (main entry for compositions; similar to the Mathematica ordering for partitions, A080577), A124734 (similar to the Abramowitz & Stegun ordering for partitions, A036036), and this sequence (similar to the Maple partition ordering, A080576), A296772.

Flatten[ Table[ Reverse[ # ] & /@ Reverse[ Sort[ Flatten[ Permutations[ # ] & /@ Partitions[ n], 1]]], {n, 6}]] (* Robert G. Wilson v, Jun 22 2005 *)

More terms from Robert G. Wilson v, Jun 22 2005

Name corrected by Andrey Zabolotskiy, May 18 2018

A129952 Binomial transform of A124625.

Original entry on oeis.org

1, 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560, 5632, 12288, 26624, 57344, 122880, 262144, 557056, 1179648, 2490368, 5242880, 11010048, 23068672, 48234496, 100663296, 209715200, 436207616, 905969664, 1879048192, 3892314112

Offset: 0

Paul Curtz, Jun 10 2007

Essentially the same as A057711: a(n) = A057711(n) for n >= 1.

Number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>2, 1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements. - Sergey Kitaev, Dec 08 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A124625, A045623, A057711, A129953 (first differences), A129954 (second differences), A129955 (third differences).

m:=15; S:=&cat[ [ 1, 2*i ]: i in [0..m] ]; [ &+[ Binomial(j-1, k-1)*S[k]: k in [1..j] ]: j in [1..2*m] ]; // Klaus Brockhaus, Jun 17 2007

LinearRecurrence[{4, -4}, {1, 1, 2, 6}, 30] (* G. C. Greubel, Jun 08 2016; corrected by Georg Fischer, Apr 02 2019 *)

{m=29; print1(1, ",", 1, ","); for(n=2, m, print1(n*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007

def A129952(n): return n<1 else 1 # Chai Wah Wu, Oct 03 2024

a(0) = 1, a(1) = 1; for n > 1, a(n) = n*2^(n-2).
G.f.: (1-3*x+2*x^2+2*x^3)/(1-2*x)^2.
E.g.f.: (1/2)*(x*exp(2*x) + x + 2). - G. C. Greubel, Jun 08 2016

Edited and extended by Klaus Brockhaus, Jun 17 2007

A159694 a(n) = 2*a(n-1) + 2^(n-1), for n > 0, with a(0)=6.

Original entry on oeis.org

6, 13, 28, 60, 128, 272, 576, 1216, 2560, 5376, 11264, 23552, 49152, 102400, 212992, 442368, 917504, 1900544, 3932160, 8126464, 16777216, 34603008, 71303168, 146800640, 301989888, 620756992, 1275068416, 2617245696, 5368709120

Offset: 0

Philippe Deléham, Apr 20 2009

Diagonal of triangles A062111, A152920.

			a(0) = 6,
a(1) = 2* 6 + 1 =  13,
a(2) = 2*13 + 2 =  28,
a(3) = 2*28 + 4 =  60,
a(4) = 2*60 + 8 = 128, ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjić and Boris Petković, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq., Vol. 17 (2014), Article 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A000079, A001787, A001792, A034007, A045623.
Cf. A045891, A062111, A111297, A152920.
Seventh row of triangle A062111. - Klaus Brockhaus, Sep 27 2009

[(12+n)*2^(n-1): n in [0..30]]; // G. C. Greubel, Sep 27 2022

Table[(6 + n/2)*2^n, {n, 0, 30}] (* Amiram Eldar, Jan 19 2021 *)

[(12+n)*2^(n-1) for n in range(30)] # G. C. Greubel, Sep 27 2022

a(n) = Sum_{k=0..n} (k+6)*binomial(n,k).
From Klaus Brockhaus, Sep 27 2009: (Start)
a(n) = (6 + n/2)*2^n.
G.f.: (6 - 11*x)/(1-2*x)^2. (End)
From Amiram Eldar, Jan 19 2021: (Start)
Sum_{n>=0} 1/a(n) = 8192*log(2) - 3934820/693.
Sum_{n>=0} (-1)^n/a(n) = 11509636/3465 - 8192*log(3/2). (End)
E.g.f.: (6 + x)*exp(2*x). - G. C. Greubel, Sep 27 2022

A188553 T(n,k) = Number of n X k binary arrays without the pattern 0 1 diagonally, vertically, antidiagonally or horizontally.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 5, 8, 7, 5, 6, 12, 12, 9, 6, 7, 17, 20, 16, 11, 7, 8, 23, 32, 28, 20, 13, 8, 9, 30, 49, 48, 36, 24, 15, 9, 10, 38, 72, 80, 64, 44, 28, 17, 10, 11, 47, 102, 129, 112, 80, 52, 32, 19, 11, 12, 57, 140, 201, 192, 144, 96, 60, 36, 21, 12, 13, 68, 187, 303, 321, 256, 176

Offset: 1

R. H. Hardin, Apr 04 2011

From Miquel A. Fiol, Feb 06 2024: (Start)
Also, T(n,k) is the number of words of length k, x(1)x(2)...x(k), on the alphabet {0,1,...,n}, such that, for i=2,...,k, x(i)=either x(i-1) or x(i)=x(i-1)-1.
For the bijection between arrays and sequences, notice that the i-th column consists of 1's and then 0's, and there are x(i)=0 to n of 1's.
Such a bijection implies that all the empirical/conjectured formulas in A188554, A188555, A188556, A188557, A188558, and A188559 become correct.
(End)

			Table starts
..2..3..4..5...6...7...8...9...10...11...12....13....14....15....16.....17
..3..5..8.12..17..23..30..38...47...57...68....80....93...107...122....138
..4..7.12.20..32..49..72.102..140..187..244...312...392...485...592....714
..5..9.16.28..48..80.129.201..303..443..630...874..1186..1578..2063...2655
..6.11.20.36..64.112.192.321..522..825.1268..1898..2772..3958..5536...7599
..7.13.24.44..80.144.256.448..769.1291.2116..3384..5282..8054.12012..17548
..8.15.28.52..96.176.320.576.1024.1793.3084..5200..8584.13866.21920..33932
..9.17.32.60.112.208.384.704.1280.2304.4097..7181.12381.20965.34831..56751
.10.19.36.68.128.240.448.832.1536.2816.5120..9217.16398.28779.49744..84575
.11.21.40.76.144.272.512.960.1792.3328.6144.11264.20481.36879.65658.115402
Some solutions for 5 X 3:
  1 1 1   1 0 0   0 0 0   1 1 1   1 1 1   1 1 1   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 1   1 1 1   1 1 1   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 1   1 0 0   1 1 0   1 1 1
  1 1 1   0 0 0   0 0 0   1 1 0   0 0 0   1 0 0   1 1 1
  1 1 1   0 0 0   0 0 0   1 0 0   0 0 0   0 0 0   1 1 0
Some solutions for T(5,3): By taking the sums of the columns in the above arrays we get 555, 100, 000, 543, 322, 432, 554. - _Miquel A. Fiol_, Feb 04 2024

R. H. Hardin, Table of n, a(n) for n = 1..9934

Diagonal is A045623.
Column 4 is A086570.
Rows 2..8 are A022856(n+4), A188554, A188555, A188556, A188557, A188558, A188559.
Upper diagonals T(n,n+i) for i=1..8 give: A001792, A001787(n+1), A000337(n+1), A045618, A045889, A034009, A055250, A055251.
Lower diagonals T(n+i,n) for i=1..7 give: A045891(n+1), A034007(n+2), A111297(n+1), A159694(n-1), A159695(n-1), A159696(n-1), A159697(n-1).
Antidiagonal sums give A065220(n+5).

T:= (n,k)-> `if`(k<=n+1, (2*n+3-k)*2^(k-2), (n+1-k)*binomial(k-1, n) * add(binomial(n, j-1)/(k-j)*T(n, j)*(-1)^(n-j), j=1..n+1)): seq(seq(T(n, 1+d-n), n=1..d), d=1..15); #Alois P. Heinz in the Sequence Fans Mailing List, Apr 04 2011 [We do not permit programs based on conjectures, but this program is now justified by Fiol's comment. - N. J. A. Sloane, Mar 09 2024]

Empirical: T(n,k) = (n+1)*2^(k-1) + (1-k)*2^(k-2) for k < n+3, and then the entire row n is a polynomial of degree n in k.
From Miquel A. Fiol, Feb 06 2024: (Start)
The above empirical formula is correct.
It can be proved that T(n,k) satisfies the recurrence
  T(n,k) = Sum_{r=1..n+1} (-1)^(r+1)*binomial(n+1,r)*T(n,k-r)
with initial values
  T(n,k) = Sum_{r=0..k-1} (n+1-r)*binomial(k-1,r) for k = 1..n+1. (End)

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