A134309 -id:A134309 - cp's OEIS frontend

A134315 A134309 * A097806.

1, 1, 1, 0, 2, 2, 0, 0, 4, 4, 0, 0, 0, 8, 8, 0, 0, 0, 0, 16, 16, 0, 0, 0, 0, 0, 32, 32, 0, 0, 0, 0, 0, 0, 64, 64

Offset: 1

Gary W. Adamson, Oct 19 2007

A134315 * [1,2,3,...] = A128135: (1, 3, 10 28, 72, 176, 416, ...).

Triangle read by rows given by [1,-1,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007

			First few rows of the triangle are:
  1;
  1, 1;
  0, 2, 2;
  0, 0, 4, 4;
  0, 0, 0, 8, 8;
  ...

Cf. A134309, A097806, A128135.

A134309 * A134315 as infinite lower triangular matrices. Triangle read by rows, for n>1, (n-1) zeros followed by 2^(n-1), 2^(n-1). As an infinite lower triangular matrix, (1, 1, 2, 4, 8, ...) in the main diagonal and (1, 2, 4, 8, ...) in the subdiagonal.

G.f.: (-1-x+x*y)/(-1+2*x*y). - R. J. Mathar, Aug 11 2015

A134310 (A000012 * A134309 + A134309 * A000012) - A000012, where the sequences are interpreted as lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 9, 11, 15, 16, 16, 17, 19, 23, 31, 32, 32, 33, 35, 39, 47, 63, 64, 64, 65, 67, 71, 79, 95, 127, 128, 128, 129, 131, 135, 143, 159, 191, 255, 256, 256, 257, 259, 263, 271, 287, 319, 383, 511

Offset: 0

Gary W. Adamson, Oct 19 2007

From M. F. Hasler, Mar 29 2022: (Start)
Both A000012 and A134309 have offset 0, so this triangular matrix also has row and column indices starting at 0.
Right (resp. left) multiplication by a diagonal matrix (such as A134309) amounts to multiplying the columns (resp. rows) of the other matrix by the diagonal elements. Therefore this matrix is the sum of the two lower triangular matrices with columns (resp. rows) filled with the same element given by sequence A134309 = (1, 1, 2, 4, 8, 16, ...), i. e., restricted to upper left 5 X 5 square:
           ( 1         )   ( 1         )   ( 1         )
           ( 1 1       )   ( 1 1       )   ( 1 1       )
  (this) = ( 1 1 2     ) + ( 2 2 2     ) - ( 1 1 1     ) .  (End)
           ( 1 1 2 4   )   ( 4 4 4 4   )   ( 1 1 1 1   )
           ( 1 1 2 4 8 )   ( 8 8 8 8 8 )   ( 1 1 1 1 1 )

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  3;
   4,  4,  5,  7;
   8,  8,  9, 11, 15;
  16, 16, 17, 19, 23, 31;
  32, 32, 33, 35, 39, 47, 63;
  ...

Cf. A000012 (all 1's), A134309 = diag(A011782 = 2^max(n-1,0), n >= 0), A000079.

Row sums are A134311.

A134310(r,c)=if(r>=c, 2^max(c-1,0)+2^max(r-1,0)-1)
matrix(8,8,i,j,A134310(i-1,j-1)) \\ M. F. Hasler, Mar 29 2022

(A000012 * A134309 + A134309 * A000012) - A000012, as infinite lower triangular matrices, where A000012 = (1; 1,1; 1,1,1; ...), and A134309 = diag(1, 1, 2, 4, 8, 16, ...) = diag(A011782 = 1 followed by 1, 2, 4, 8, ... = powers of 2).

Row sums: A134311 = (1, 2, 7, 20, 51, 122, 281, 632, ...).

Edited and offset corrected to 0 by M. F. Hasler, Mar 29 2022

A134317 Triangle, A128174 * A134309 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 4, 1, 0, 2, 0, 8, 0, 1, 0, 4, 0, 16, 1, 0, 2, 0, 8, 0, 32, 0, 1, 0, 4, 0, 16, 0, 64, 1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 1, 0, 4, 0, 16, 0, 64, 0, 256

Offset: 1

Gary W. Adamson, Oct 19 2007

Is this the same as A123641? - R. J. Mathar, Mar 28 2012

			First few rows of the triangle are:
1;
0, 1;
1, 0, 2;
0, 1, 0, 4;
1, 0, 2, 0, 8;
0, 1, 0, 4, 0, 16;
1, 0, 2, 0, 8, 0, 32;
0, 1, 0, 4, 0, 16, 0, 64;
...

Cf. A128174, A134309, A001045 (row sums).

T(n,k) = 0 if n+k odd, else T(n,1) =1, T(n,k)=2^(k-2) if k>=2. - R. J. Mathar, Sep 01 2024

A134312 A097806 * A134309.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 2, 4, 0, 0, 0, 4, 8, 0, 0, 0, 0, 8, 16, 0, 0, 0, 0, 0, 16, 32, 0, 0, 0, 0, 0, 0, 32, 64

Offset: 0

Gary W. Adamson, Oct 19 2007

A134313 = A134309 * A097806.

			First few rows of the triangle:
  1;
  1, 1;
  0, 1, 2;
  0, 0, 2, 4;
  0, 0, 0, 4, 8;
  0, 0, 0, 0, 8, 16;
  ...

Cf. A134309, A097806, A134313.

A097806 * A134309 as infinite lower triangular matrices, where A097806 = the pairwise operator. Infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main and subdiagonals.

A143038 Triangle read by rows, A000012 * A134309 * A000012, where A134309 = an infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 6, 7, 8, 8, 12, 14, 15, 16, 16, 24, 28, 30, 31, 32, 32, 48, 56, 60, 62, 63, 64, 64, 96, 112, 120, 124, 126, 127, 128, 128, 192, 224, 240, 248, 252, 254, 255, 256, 256, 384, 448, 480, 496, 504, 508, 510, 511, 512

Offset: 0

Gary W. Adamson, Jul 18 2008

Row sums = A002064, the Cullen numbers (1, 3, 9, 25, 65, 161, 385, ...).

			First few rows of the triangle:
   1;
   1,  2;
   2,  3,  4;
   4,  6 , 7,  8;
   8, 12, 14, 15, 16;
  16, 24, 28, 30, 21, 32;
  32, 48, 56, 60, 62, 63, 64;
  ...

Cf. A134309, A002064.

Cf. A002262, A003056, A023532.

a(n) = 2^i - 2^(i-1-j)*[jA003056(n), j = A002262(n). - Yuchun Ji, May 15 2020

A081294 Expansion of (1-2x)/(1-4x).

Original entry on oeis.org

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832

Offset: 0

Paul Barry, Mar 17 2003

Binomial transform of A046717. Second binomial transform of A000302 (with interpolated zeros). Partial sums are A007583.
Counts closed walks of length 2n at a vertex of the cyclic graph on 4 nodes C_4. With interpolated zeros, counts closed walks of length n at a vertex of the cyclic graph on 4 nodes C_4. - Paul Barry, Mar 10 2004
In general, Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*r^j has expansion (1 - (r+1)*x)/(1 - (r+3)*x - (r-1)*(r+3)*x^2 + (r-1)^3*x^3). - Paul Barry, Jun 04 2005 [corrected by Jason Yuen, Jan 20 2025]
a(n) is the number of binary strings of length 2n with an even number of 0's (and hence an even number of 1's). - Toby Gottfried, Mar 22 2010
Number of compositions of n where there are 2 sorts of part 1, 4 sorts of part 2, 8 sorts of part 3, ..., 2^k sorts of part k. - Joerg Arndt, Aug 04 2014
a(n) is also the number of permutations simultaneously avoiding 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
INVERT transform of powers of 2 (A000079). - Alois P. Heinz, Feb 11 2021
a(n) is the number of elements in an n-interval of the binomial poset of even-sized subsets of positive integers, cf. Stanley reference and second formula by Paul Barry.  Each multichain 0 = x_0 <= x_1 <= x_2 = 1 in such an n-interval corresponds to a closed walk described above by Paul Barry.  More generally, each multichain 0 = x_0 <= x_1 <= ... <= x_k = 1 corresponds to a closed walk of length 2n on the k-dimensional hypercube, cf. A054879, A092812, A121822. - Geoffrey Critzer, Apr 21 2023

			G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 128*x^4 + 512*x^5 + 2048*x^6 + 8192*x^7 + ...

Richard P. Stanley, Enumerative Combinatorics, Vol 1, second edition, Example 3.18.3-f, page 323.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (4).

Row sums of triangle A136158.

Cf. A000079, A081295, A009117, A016742, A054879, A092812, A121822. Essentially the same as A004171.

[(4^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Jul 26 2011

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-2*x)/(1-4*x))); // Marius A. Burtea, Jan 20 2020

a:= n-> 2^max(0, (2*n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 20 2017

CoefficientList[Series[(1-2x)/(1-4x),{x,0,40}],x] (* or *)
Join[{1}, NestList[4 # &, 2, 40]] (* Harvey P. Dale, Apr 22 2011 *)

a(n)=1<Charles R Greathouse IV, Jul 25 2011

x='x+O('x^100); Vec((1-2*x)/(1-4*x)) \\ Altug Alkan, Dec 21 2015

G.f.: (1-2*x)/(1-4*x).
a(n) = 4*a(n-1) n > 1, with a(0)=1, a(1)=2.
a(n) = (4^n+0^n)/2 (i.e., 1 followed by 4^n/2, n > 0).
E.g.f.: exp(2*x)*cosh(2*x) = (exp(4*x)+exp(0))/2. - Paul Barry, May 10 2003
a(n) = Sum_{k=0..n} C(2*n, 2*k). - Paul Barry, May 20 2003
a(n) = A001045(2*n+1) - A001045(2*n-1) + 0^n/2. - Paul Barry, Mar 10 2004
a(n) = 2^n*A011782(n); a(n) = gcd(A011782(2n), A011782(2n+1)). - Paul Barry, Jan 12 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j). - Paul Barry, Jun 04 2005
a(n) = Sum_{k=0..n} A038763(n,k). - Philippe Deléham, Sep 22 2006
a(n) = Integral_{x=0..4} p(n,x)^2/(Pi*sqrt(x(4-x))) dx, where p(n,x) is the sequence of orthogonal polynomials defined by C(2*n,n): p(n,x) = (2*x-4)*p(n-1,x) - 4*p(n-2,x), with p(0,x)=1, p(1,x)=-2+x. - Paul Barry, Mar 01 2007
a(n) = ((2+sqrt(4))^n + (2-sqrt(4))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008
a(n) = A000079(n) * A011782(n). - Philippe Deléham, Dec 01 2008
a(n) = A004171(n-1) = A028403(n) - A000079(n) for n >= 1. - Jaroslav Krizek, Jul 27 2009
a(n) = Sum_{k=0..n} A201730(n,k)*3^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..n} A134309(n,k)*2^k = Sum_{k=0..n} A055372(n,k). - Philippe Deléham, Feb 04 2012
G.f.: Q(0), where Q(k) = 1 - 2*x/(1 - 2/(2 - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
E.g.f.: 1/2 + exp(4*x)/2 = (Q(0)+1)/2, where Q(k) = 1 + 4*x/(2*k+1 - 2*x*(2*k+1)/(2*x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
a(n) = ceiling( 2^(2n-1) ). - Wesley Ivan Hurt, Jun 30 2013
G.f.: 1 + 2*x/(1 + x)*( 1 + 5*x/(1 + 4*x)*( 1 + 8*x/(1 + 7*x)*( 1 + 11*x/(1 + 10*x)*( 1 + ... )))). - Peter Bala, May 27 2017
Sum_{n>=0} 1/a(n) = 5/3. - Amiram Eldar, Aug 18 2022
Sum_{n>=0} a(n)*x^n/A000680(n) = E(x)^2 where E(x) = Sum_{n>=0} x^n/A000680(n). - Geoffrey Critzer, Apr 21 2023

A093143 Expansion of (1-5x)/(1-10x).

Original entry on oeis.org

1, 5, 50, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000, 5000000000000000000, 50000000000000000000, 500000000000000000000

Offset: 0

Paul Barry, Mar 24 2004

Partial sums are A093142. A convex combination of 10^n and 0^n.
a(n) is the number of compositions of even natural numbers in n parts <= 9 (0 is counted as a part); also the number of ways of placing of an even number of indistinguishable objects into n distinguishable boxes with the condition that at most 9 objects can be placed in each box. - Adi Dani, May 17 2011
See an A246057 comment with a reference for the k-family satisfying a so-called curious cubic identity involving A246057(k-1), a(k) and A002277(k). - Wolfdieter Lang, Feb 07 2017

			From _Adi Dani_, May 17 2011: (Start)
a(2)=50: there are 50 compositions of even numbers into 2 parts <= 9:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(1,9),(9,1),(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
(3,9),(9,3),(4,8),(8,4),(5,7),(7,5),(6,6);
(5,9),(9,5),(6,8),(8,6),(7,7);
(7,9),(9,7),(8,8);
(9,9).
(End)
Curious cubic identities (see a comment above): 1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ... - _Wolfdieter Lang_, Feb 07 2017

Adi Dani, Restricted compositions of natural numbers
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A002277, A055372, A093142, A134309, A246057.

Table[Ceiling[1/2*10^n],{n,0,30}] (* Adi Dani, Jun 20 2011 *)
Join[{1},NestList[10#&,5,20]] (* Harvey P. Dale, Apr 10 2021 *)

Vec((1-5*x)/(1-10*x) + O(x^100)) \\ Altug Alkan, Nov 01 2015

a(n) = 5*10^n/10 for n > 0.
a(n) = Sum_{k=0..n} A134309(n,k)*5^k = Sum_{k=0..n} A055372(n,k)*4^k. - Philippe Deléham, Feb 04 2012
From Elmo R. Oliveira, Aug 21 2024: (Start)
E.g.f.: (exp(10*x) + 1)/2.
a(n) = 10*a(n-1) for n > 1. (End)

a(19)-a(21) from Elmo R. Oliveira, Aug 21 2024

A081341 Expansion of exp(3x)cosh(3*x).

Original entry on oeis.org

1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A081340. 3rd binomial transform of (1,0,9,0,81,0,729,0,...).
For m > 1, n > 0, A166469(A002110(m)*a(n)) = (n+1)*A000045(m+1). For n > 0, A166469(a(n)) = 2n. - Matthew Vandermast, Nov 05 2009
Number of compositions of even natural numbers in n parts <= 5. - Adi Dani, May 29 2011

			From _Adi Dani_, May 29 2011: (Start)
a(2)=18: there are 18 compositions of even natural numbers into 2 parts <= 5:
  for 0: (0,0);
  for 2: (0,2),(2,0),(1,1);
  for 4: (0,4),(4,0),(1,3),(3,1),(2,2);
  for 6: (1,5),(5,1),(2,4),(4,2),(3,3);
  for 8: (3,5),(5,3),(4,4);
  for 10: (5,5).  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..125
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A000244, A034494, A081340, A081342.

a:= proc(n) option remember; `if`(n=0, 1,
      add(3^j*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2017

Table[Ceiling[1/2(6^n)], {n, 0, 25}]
CoefficientList[Series[-(-1 + 3 x)/(1 - 6 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1},NestList[6#&,3,30]] (* Harvey P. Dale, May 25 2019 *)

x='x+O('x^66); /* that many terms */
Vec((1-3*x)/(1-6*x)) /* show terms */ /* Joerg Arndt, May 29 2011 */

a(0)=1, a(n) = 6^n/2, n > 0.
G.f.: (1-3*x)/(1-6*x).
E.g.f.: exp(3*x)*cosh(3*x).
a(n) = A000244(n)*A011782(n). - Philippe Deléham, Dec 01 2008
a(n) = ((3+sqrt(9))^n + (3-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
a(n) = Sum_{k=0..n} A134309(n,k)*3^k = Sum_{k=0..n} A055372(n,k)*2^k. - Philippe Deléham, Feb 04 2012
From Sergei N. Gladkovskii, Jul 19 2012: (Start)
a(n) = ((8*n-4)*a(n-1) - 12*(n-2)*a(n-2))/n, a(0)=1, a(1)=3.
E.g.f. (exp(6*x) + 1)/2 = 1 + 3*x/(G(0) - 6*x) where G(k) = 6*x + 1 + k - 6*x*(k+1)/G(k+1) (continued fraction, Euler's 1st kind, 1-step). (End)
"INVERT" transform of A000244. - Alois P. Heinz, Sep 22 2017

Typo in A-number fixed by Klaus Brockhaus, Apr 04 2010

A055372 Invert transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 4, 12, 12, 4, 8, 32, 48, 32, 8, 16, 80, 160, 160, 80, 16, 32, 192, 480, 640, 480, 192, 32, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 128, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 256, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256

Offset: 0

Christian G. Bower, May 16 2000

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

T(n,k) is the number of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence. For example, T(2,1)=4 because we have {(01)},{(10)},{(0),(1)},{(1),(0)}. - Geoffrey Critzer, Apr 06 2013

			Triangle begins:
  1;
  1,  1;
  2,  4,  2;
  4, 12, 12,  4;
  8, 32, 48, 32,  8;
  ...

N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A081294. Cf. A000079, A007318, A055373, A055374.
Cf. A134309.
T(2n,n) gives A098402.

nn=10;f[list_]:=Select[list,#>0&];a=(x+y x)/(1-(x+y x));Map[f,CoefficientList[Series[1/(1-a),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Apr 06 2013 *)

a(n,k) = 2^(n-1)*C(n, k), for n>0.
G.f.: A(x, y)=(1-x-xy)/(1-2x-2xy).
As an infinite lower triangular matrix, equals A134309 * A007318. - Gary W. Adamson, Oct 19 2007
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Feb 05 2012

A082137 Square array of transforms of binomial coefficients, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 8, 1, 5, 20, 40, 40, 16, 1, 6, 30, 80, 120, 96, 32, 1, 7, 42, 140, 280, 336, 224, 64, 1, 8, 56, 224, 560, 896, 896, 512, 128, 1, 9, 72, 336, 1008, 2016, 2688, 2304, 1152, 256, 1, 10, 90, 480, 1680, 4032, 6720, 7680, 5760, 2560, 512

Offset: 0

Paul Barry, Apr 06 2003

Rows are associated with the expansions of (x^k/k!)exp(x)cosh(x) (leading zeros dropped). Rows include A011782, A057711, A080929, A082138, A080951, A082139, A082140, A082141. Columns are of the form 2^(k-1)C(n+k, k). Diagonals include A069723, A082143, A082144, A082145, A069720.
T(n, k) is also the number of idempotent order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha)= |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
Read as a triangle this is A119468 with rows reversed. A119468 has e.g.f. exp(z*x)/(1-tanh(x)). - Peter Luschny, Aug 01 2012
Read as a triangle this is a subtriangle of A198793. - Philippe Deléham, Nov 10 2013

			Rows begin
  1 1  2   4   8 ...
  1 2  6  16  40 ...
  1 3 12  40 120 ...
  1 4 20  80 280 ...
  1 5 30 140 560 ...
Read as a triangle, this begins:
  1
  1, 1
  1, 2,  2
  1, 3,  6,  4
  1, 4, 12, 16,   8
  1, 5, 20, 40,  40, 16
  1, 6, 30, 80, 120, 96, 32
  ... - _Philippe Deléham_, Nov 10 2013

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.

Cf. A119468, A007318, A134309.

# As a triangular array:
T := (n,k) -> 2^(k+0^k-1)*binomial(n,k):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Nov 10 2017

rows = 11; t[n_, k_] := 2^(n-1)*(n+k)!/(n!*k!); t[0, ] = 1; tkn = Table[ t[n, k], {k, 0, rows}, {n, 0, rows}]; Flatten[ Table[ tkn[[ n-k+1, k ]], {n, 1, rows}, {k, 1, n}]] (* _Jean-François Alcover, Jan 20 2012 *)

def A082137_row(n) : # as a triangular array
    var('z')
    s = (exp(z*x)/(1-tanh(x))).series(x,n+2)
    t = factorial(n)*s.coefficient(x,n)
    return [t.coefficient(z,n-k) for k in (0..n)]
for n in (0..7) : print(A082137_row(n))  # Peter Luschny, Aug 01 2012

Square array defined by T(n, k)=(2^(n-1)+0^n/2)C(n + k, n)= Sum{k=0..n, C(n+k, k+j)C(k+j, k)(1+(-1)^j)/2 }.
As an infinite lower triangular matrix, equals A007318 * A134309. - Gary W. Adamson, Oct 19 2007
O.g.f. for array read as a triangle: (1-x*(1+t))/((1-x)*(1-x*(1+2*t))) = 1 + x*(1+t) + x^2*(1+2*t+2*t^2) + x^3*(1+3*t+6*t^2+4*t^3) + .... - Peter Bala, Apr 26 2012
For array read as a triangle: T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) -2*T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 10 2013

cp's OEIS Frontend

A134315 A134309 * A097806.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A134310 (A000012 * A134309 + A134309 * A000012) - A000012, where the sequences are interpreted as lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A134317 Triangle, A128174 * A134309 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A134312 A097806 * A134309.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A143038 Triangle read by rows, A000012 * A134309 * A000012, where A134309 = an infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A081294 Expansion of (1-2*x)/(1-4*x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

PARI

Formula

A093143 Expansion of (1-5*x)/(1-10*x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A081341 Expansion of exp(3*x)*cosh(3*x).

Views

Author

Keywords

Comments

Examples

A081294 Expansion of (1-2x)/(1-4x).

A093143 Expansion of (1-5x)/(1-10x).

A081341 Expansion of exp(3x)cosh(3*x).