A093178 -id:A093178 - cp's OEIS frontend

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A161738 Sequence related to the column sums of the BG2 matrix.

Original entry on oeis.org

1, 1, 3, 15, 35, 315, 693, 9009, 19305, 328185, 692835, 14549535, 30421755, 760543875, 1579591125, 45808142625, 94670161425, 3124115327025, 6432002143875, 237984079323375, 488493636505875, 20028239096740875

Offset: 1

Johannes W. Meijer, Jun 18 2009

G. C. Greubel, Table of n, a(n) for n = 1..667

Cf. A161736, A005408 and A093178.

[1] cat [(&*[2*n-2*k-3:k in [0..Floor(n/2 -1)]]): n in [2..50]]; // G. C. Greubel, Sep 26 2018

Table[Product[(2*n - 3 - 2*k), {k, 0, Floor[n/2 - 1]}], {n, 1, 50}] (* G. C. Greubel, Sep 26 2018 *)

for(n=1,50, print1(prod(k=0,floor(n/2 -1), 2*n-2*k-3), ", ")) \\ G. C. Greubel, Sep 26 2018

a(n) = product((2*n-3-2*k), k=0..floor(n/2-1)).
numer(a(n+2)/a(n+1)) = A005408(n) for n=>0.
denom(a(n+2)/a(n+1)) = A093178(n) for n=>0.

A289296 a(n) = (n - 1)(2floor(n/2) + 1).

Original entry on oeis.org

-1, 0, 3, 6, 15, 20, 35, 42, 63, 72, 99, 110, 143, 156, 195, 210, 255, 272, 323, 342, 399, 420, 483, 506, 575, 600, 675, 702, 783, 812, 899, 930, 1023, 1056, 1155, 1190, 1295, 1332, 1443, 1482, 1599, 1640, 1763, 1806, 1935, 1980, 2115, 2162, 2303, 2352, 2499, 2550, 2703, 2756, 2915

Offset: 0

Jean-François Alcover and Paul Curtz, Jul 02 2017

Summing a(n) by pairs, one gets -1, 9, 35, 77, 135, ... = A033566.

A198442(k) is a member of this sequence if k == 0 or 1 (mod 4). - Bruno Berselli, Jul 04 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Subsequence of A214297.

Cf. A023443, A033566, A093178, A109613.

Table[(n - 1) (2 Floor[n/2] + 1), {n, 0, 60}] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {-1, 0, 3, 6, 15}, 61]

a(n)=(n\2*2+1)*(n-1) \\ Charles R Greathouse IV, Jul 02 2017

Vec(-(1 - x - 5*x^2 - x^3 - 2*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^60)) \\ Colin Barker, Jul 02 2017

def A289296(n): return (n-1)*(n|1) # Chai Wah Wu, Jan 18 2023

a(n) = A023443(n) * A109613(n).
a(n) = n^2-1 if n is even and n^2-n if n is odd.
n^2 - a(n) = A093178(n).
From Colin Barker, Jul 02 2017: (Start)
G.f.: -(1 - x - 5*x^2 - x^3 - 2*x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4. (End)

A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2n-5)a(n-1).

Original entry on oeis.org

1, -1, 2, -6, 30, -210, 1890, -20790, 270270, -4054050, 68918850, -1309458150, 27498621150, -632468286450, 15811707161250, -426916093353750, 12380566707258750, -383797567925021250, 12665319741525701250

Offset: 1

Paul Curtz, May 17 2007

sign

Sequence is also the first column of the inverse of the infinite lower triangular matrix M, where M(j,k) = 1+2*(k-1)*(j-k) for k < j, M(j,k) = 1 for k = j, M(j,k) = 0 for k > j.
Upper left 6 X 6 submatrix of M is
  [ 1  0  0  0  0  0 ]
  [ 1  1  0  0  0  0 ]
  [ 1  3  1  0  0  0 ]
  [ 1  5  5  1  0  0 ]
  [ 1  7  9  7  1  0 ]
  [ 1  9 13 13  9  1 ],
and upper left 6 X 6 submatrix of M^-1 is
  [    1    0    0    0    0    0 ]
  [   -1    1    0    0    0    0 ]
  [    2   -3    1    0    0    0 ]
  [   -6   10   -5    1    0    0 ]
  [   30  -50   26   -7    1    0 ]
  [ -210  350 -182   50   -9    1 ].
Row sums of M are 1, 2, 5, 12, 25, 46, ... (see A116731); diagonal sums of M are 1, 1, 2, 4, 7, 13, 20, 32, 45, 65, 86, 116, 147, 189, ... with first differences 0, 1, 2, 3, 6, 7, 12, 13, 20, 21, 30, 31, 42, ... and second differences 1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, ... (see A093178).

G. C. Greubel, Table of n, a(n) for n = 1..400

Cf. A093178, A097801, A116731.

F:=Factorial;; Concatenation([1,-1], List([3..25], n-> (-1)^(n+1)*F(2*n-5)/(2^(n-4)*F(n-3)) )); # G. C. Greubel, Nov 25 2019

m:=19; M:=Matrix(IntegerRing(), m, m, [< j, k, Maximum(0, 1+2*(k-1)*(j-k)) > : j, k in [1..m] ] ); Transpose(ColumnSubmatrix(M^-1, 1, 1)); // Klaus Brockhaus, May 21 2007

F:=Factorial; [1,-1] cat [(-1)^(n+1)*F(2*n-5)/(2^(n-4)*F(n-3)): n in [3..25]]; // G. C. Greubel, Nov 25 2019

seq(`if`(n<3, (-1)^(n-1), (-1)^(n-1)*(2*n-5)!/(2^(n-4)*(n-3)!)), n=1..25); # G. C. Greubel, Nov 25 2019

a[n_]:= -(2*n-5)*a[n-1]; a[1]=1; a[2]=-1; a[3]=2; Array[a, 20] (* Robert G. Wilson v *)
Table[If[n<3, (-1)^(n-1), (-1)^(n+1)*(2*n-5)!/(2^(n-4)*(n-3)!)], {n,25}] (* G. C. Greubel, Nov 25 2019 *)

{m=19; print1(1, ",", -1, ","); print1(a=2, ","); for(n=4, m, k=-(2*n-5)*a; print1(k, ","); a=k)} \\ Klaus Brockhaus, May 21 2007

{print1(1, ",", -1, ","); for(n=3, 19, print1((-1)^(n-1)*(2*(n-2))!/((n-2)!*2^(n-3)), ","))} \\ Klaus Brockhaus, May 21 2007

{m=19; M=matrix(m, m, j, k, if(k>j, 0, if(k==j, 1, 1+2*(k-1)*(j-k)))); print((M^-1)[, 1]~)} \\ Klaus Brockhaus, May 21 2007

f=factorial; [1,-1]+[(-1)^(n+1)*f(2*n-5)/(2^(n-4)*f(n-3)) for n in (3..25)] # G. C. Greubel, Nov 25 2019

a(n) = (-1)^(n-1)*A097801(n-2) = (-1)^(n-1)*(2*(n-2))!/((n-2)!*2^(n-3)) for n > 2, with a(1)=1, a(2)=-1.

G.f.: 1 + x - x*W(0) , where W(k) = 1 + 1/( 1 - x*(2*k+1)/( x*(2*k+1) - 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 22 2013

Edited and extended by Klaus Brockhaus and Robert G. Wilson v, May 21 2007

A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 6, 3, 2, 1, 5, 5, 1, 3, 5, 4, 4, 4, 2, 2, 1, 7, 6, 8, 4, 3, 5, 7, 9, 7, 6, 5, 4, 3, 2, 1, 8, 11, 7, 11, 1, 4, 5, 7, 9, 10, 9, 5, 7, 6, 2, 4, 3, 2, 1, 15, 10, 9, 9, 14, 6, 3, 5, 7, 9, 11, 12, 8, 18, 8, 8, 7, 6, 4, 4, 3, 2, 1, 13, 12, 11, 10, 12, 17, 1, 6, 5, 7, 9, 11, 13, 14, 13, 16, 6, 10, 9, 8, 2, 6, 5, 4, 3, 2, 1

Offset: 1

Boris Putievskiy, Aug 29 2024

A208233 presents an algorithm for generating permutations, where each generated permutation is self-inverse.

The sequence in each column k possesses two properties: it is both a self-inverse permutation and an intra-block permutation of natural numbers.

			Table begins:
    k=    1   2   3   4   5   6
  -----------------------------------
  n= 1:   1,  1,  3,  1,  5,  1, ...
  n= 2:   2,  2,  2,  3,  2,  5, ...
  n= 3:   3,  3,  1,  2,  3,  3, ...
  n= 4:   6,  5,  4,  4,  4,  4, ...
  n= 5:   5,  4,  8,  5,  1,  2, ...
  n= 6:   4,  6,  6, 11,  6,  6, ...
  n= 7:   7,  7,  7,  7, 14,  7, ...
  n= 8:   9, 11,  5,  9,  8, 17, ...
  n= 9:   8,  9,  9,  8, 12,  9, ...
  n= 10: 10, 10, 18, 10, 10, 15, ...
  n= 11: 15,  8, 11,  6, 11, 11, ...
  n= 12: 12, 12, 16, 12,  9, 13, ...
  n= 13: 13, 13, 13, 13, 13, 12, ...
  n= 14: 14, 19, 14, 23,  7, 14, ...
  n= 15: 11, 15, 15, 15, 15, 10, ...
  n= 16: 16, 17, 12, 21, 30, 16, ...
  n= 17: 20, 16, 17, 17, 17,  8, ...
  n= 18: 18, 18, 10, 19, 28, 18, ...
     ... .
In column 3, the first 3 blocks have lengths 3,6 and 9. In column 6, the first 2 blocks have lengths 6 and 12. Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
  1;
  2,1;
  3,2,3;
  6,3,2,1;
  5,5,1,3,5;
  4,4,4,2,2,1;

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000194, A002024, A002260, A003991, A062707, A074294, A093178, A111651, A124625, A188568, A208233, A371355.

T[n_,k_]:=Module[{L,R,P,result},L=Ceiling[(Sqrt[8*n*k+k^2]-k)/(2*k)]; R=n-k*(L-1)*L/2; P=(((-1)^Max[R,k*L+1-R]+1)*R-((-1)^Max[R,k*L+1-R]-1)*(k*L+1-R))/2; result=P+k*(L-1)*L/2]
Nmax=18; Table[T[n,k],{n,1,Nmax},{k,1,Nmax}]

T(n,k) = P(n,k) + k*(L(n,k)-1)*L(n,k)/2 = P(n,k) + A062707(L(n-1),k), where L(n,k) = ceiling((sqrt(8*n*k+k^2)-k)/(2*k)), R(n,k) = n-k*(L(n,k)-1)*L(n,k)/2, P(n,k) = (((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))+1)*R(n,k)-((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))-1)*(k*L(n,k)+1-R(n,k)))/2.

T(n,1) = A188568(n). T(1,k) = A093178(k). T(n,n) = A124625(n). L(n,1) = A002024(n). L(n,2) = A000194(n). L(n,3) = A111651(n). L(n,4) = A371355(n). R(n,1) = A002260(n). R(n,2) = A074294(n).

A062557 2n-1 1's followed by a 2.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Jason Earls, Jul 02 2001

Is the number .12111211111211111112... irrational?

Cf. A005369, A093178.

[1+Floor(Sqrt(n+2)+1/2)-Floor(Sqrt(n+1)+1/2) : n in [0..100]]; // Wesley Ivan Hurt, May 26 2015

A062557:=n->1+floor(sqrt(n+2)+1/2)-floor(sqrt(n+1)+1/2): seq(A062557(n), n=0..100); # Wesley Ivan Hurt, May 26 2015

Table[1 + Floor[Sqrt[n + 2] + 1/2] - Floor[Sqrt[n + 1] + 1/2], {n, 0,
100}] (* Wesley Ivan Hurt, May 26 2015 *)

v=[]; for(n=0,200,v=concat(v,1+issquare(5+4*n))); v

a(n) = 1 + A005369(n+1). - Wesley Ivan Hurt, May 26 2015

A226725 Denominator of the median of {1, 1/2, 1/3, ..., 1/n}.

Original entry on oeis.org

1, 4, 2, 12, 3, 24, 4, 40, 5, 60, 6, 84, 7, 112, 8, 144, 9, 180, 10, 220, 11, 264, 12, 312, 13, 364, 14, 420, 15, 480, 16, 544, 17, 612, 18, 684, 19, 760, 20, 840, 21, 924, 22, 1012, 23, 1104, 24, 1200, 25, 1300, 26, 1404, 27, 1512, 28, 1624, 29, 1740, 30

Offset: 1

Clark Kimberling, Jun 19 2013

			median{1, 1/2, 1/3, 1/4} = (1/2 + 1/3)/2 = 7/12, so that a(4) = 12.

Clark Kimberling, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A093178 (numerators), A061579.

A226725:=n->n^(1/2 + (-1)^n/2)*(n + 2^(1/2 + (-1)^n/2))/2: seq(A226725(n), n=1..100); # Wesley Ivan Hurt, Feb 27 2015

Denominator[Table[Median[Table[1/k, {k, n}]], {n, 120}]]
f[n_] := If[ OddQ@ n, Floor[(n + 1)/2], n(n/2 + 1)]; Array[f, 59] (* Robert G. Wilson v, Feb 27 2015 *)
With[{nn=30},Riffle[Range[nn],Table[2n+2n^2,{n,nn}]]] (* Harvey P. Dale, May 26 2019 *)
Riffle[Range[60],LinearRecurrence[{3,-3,1},{4,12,24},60]] (* Harvey P. Dale, Oct 03 2023 *)

Vec(x*(x^2-4*x-1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Feb 27 2015

a(n) = (n+1)/2 if n is odd, a(n) = n*(n/2+1) if n is even.
G.f.: W(0), where W(k)= 1 + 2*x*(k+2)/( 1 - x/(x + 2*(k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). - Colin Barker, Feb 27 2015
G.f.: x*(x^2-4*x-1) / ((x-1)^3*(x+1)^3). - Colin Barker, Feb 27 2015
a(n) = n^(1/2 + (-1)^n/2)*(n + 2^(1/2 + (-1)^n/2))/2. - Wesley Ivan Hurt, Feb 27 2015
a(n) = Sum_{k=0..n} (-1)^k * A061579(n,k). - Alois P. Heinz, Feb 10 2023

Formula changed for even terms by Luca Brigada Villa, Jun 20 2013

A110491 Expansion of e.g.f.: sqrt(1+2x)/sqrt(1-2x).

Original entry on oeis.org

1, 2, 4, 24, 144, 1440, 14400, 201600, 2822400, 50803200, 914457600, 20118067200, 442597478400, 11507534438400, 299195895398400, 8975876861952000, 269276305858560000, 9155394399191040000, 311283409572495360000

Offset: 0

Paul Barry, Jul 22 2005

Row sums of exponential Riordan array [1, arctanh(2x)]. - Paul Barry, Apr 17 2008

Conjecture: {a(n-1), n>=1} is the T-transform of A093178, where T maps a sequence {b(n), n>=1} to the sequence {c(n)} defined by c(n) = det(M_n), where M_n is the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j. - Lechoslaw Ratajczak, Aug 04 2021

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

Cf. A000246, A002867, A063886.

S:= series(sqrt(1+2*x)/sqrt(1-2*x),x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, Jun 08 2016

With[{nn=20},CoefficientList[Series[Sqrt[1+2x]/Sqrt[1-2x],{x,0,nn}],x] Range[0,nn]!] (* or *) Join[{1},Table[2n!Binomial[n-1,Floor[(n-1)/2]], {n,20}]] (* Harvey P. Dale, Nov 11 2011 *)
Table[2^n Binomial[1/2,n] n! Hypergeometric2F1[1/2, -n, 3/2 - n, -1], {n, 0, 20}] (* Benedict W. J. Irwin, Jun 06 2016 *)

my(x='x+O('x^25)); Vec(serlaplace(sqrt(1+2*x)/sqrt(1-2*x))) \\ Michel Marcus, Aug 05 2021

E.g.f.: sqrt((1+2x)/(1-2x)); a(n)=2*n!*binomial(n-1, floor((n-1)/2))+0^n.
The sequence 0,1,0,2,0,4,... has e.g.f. arctanh(x). - Paul Barry, Apr 17 2008
D-finite with recurrence a(n) -2*a(n-1) -4*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Sep 20 2012
a(n) ~ 2^(n+1)*n^n/exp(n). - Vaclav Kotesovec, Sep 25 2013
a(n) = 2^n*binomial(1/2,n)*n!*2F1(1/2,-n;3/2-n;-1). - Benedict W. J. Irwin, Jun 06 2016
From Robert Israel, Jun 08 2016: (Start)
a(n) = n! * A063886(n).
E.g.f. satisfies 2*g(x)+(4*x^2-1)*g'(x) = 0, from which Mathar's recurrence follows. (End)
Sum_{n>=0} 1/a(n) = 1 + (StruveL(-1,1/2) + StruveL(0,1/2))*Pi/4, where StruveL is the modified Struve function. - Amiram Eldar, Aug 15 2025

A131352 Row sums of triangle A133935.

Original entry on oeis.org

1, 2, 6, 14, 32, 72, 160, 352, 768, 1664, 3584, 7680, 16384, 34816, 73728, 155648, 327680, 688128, 1441792, 3014656, 6291456, 13107200, 27262976, 56623104, 117440512, 243269632, 503316480, 1040187392, 2147483648, 4429185024

Offset: 0

Gary W. Adamson, Sep 29 2007

			a(3) = 14 = sum of row 3 terms of triangle A133935: (1 + 3 + 9 + 1); = (1, 3, 3, 1) dot (1, 1, 3, 1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A133935, A093178.

CoefficientList[Series[(1-2x+2x^2-2x^3)/(1-2x)^2,{x,0,40}],x] (* or *) LinearRecurrence[{4,-4},{1,2,6,14},40] (* Harvey P. Dale, Dec 04 2021 *)

Binomial transform of A093178: (1, 1, 3, 1, 5, 1, 7, 1...)
a(n) = A129954(n), n>1. G.f.: (1-2x+2x^2-2x^3)/(1-2x)^2. [R. J. Mathar, Dec 13 2008]
a(n) = 2^(n-2)*(n+4) for n>1. - _Colin Barker, Jun 05 2012

Extended by R. J. Mathar, Dec 13 2008

A171232 Array read by antidiagonals, T(n,k) = 2*(n/k) - 1, if n mod k = 0; otherwise, T(n,k) = 1.

Original entry on oeis.org

1, 3, 1, 5, 1, 1, 7, 1, 1, 1, 9, 3, 1, 1, 1, 11, 1, 1, 1, 1, 1, 13, 5, 1, 1, 1, 1, 1, 15, 1, 3, 1, 1, 1, 1, 1, 17, 7, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 9, 5, 3, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 11, 1, 1, 1, 1

Offset: 1

Ross La Haye, Dec 05 2009

T(n,1): continued fraction expansion of coth(1).

T(n,2): continued fraction expansion of tan(1) = cot(pi/2 - 1).

			Array begins
1 1 1 1 1 ...
3 1 1 1 1 ...
5 1 1 1 1 ...
7 3 1 1 1 ...
9 1 1 1 1 ...
.............

Cf. T(n, 1) = A005408(n-1), T(n, 2) = A093178(n-1), A171233, A077049.

T[n_,k_] := If[Divisible[n, k], 2*(n/k) - 1, 1]; Table[T[n-k+1, k], {n, 1, 10}, {k,1, n}] //Flatten (* Amiram Eldar, Jun 29 2020 *)

T(n,k) = A171233(n,k) - A077049(n,k).

More terms from Amiram Eldar, Jun 29 2020

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cp's OEIS Frontend

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A161738 Sequence related to the column sums of the BG2 matrix.

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A289296 a(n) = (n - 1)*(2*floor(n/2) + 1).

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A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2*n-5)*a(n-1).

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A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

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A062557 2n-1 1's followed by a 2.

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A226725 Denominator of the median of {1, 1/2, 1/3, ..., 1/n}.

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A289296 a(n) = (n - 1)(2floor(n/2) + 1).

A129779 a(1) = 1, a(2) = -1, a(3) = 2; for n > 3, a(n) = -(2n-5)a(n-1).