A054977 -id:A054977 - cp's OEIS frontend

A237448 Square array T(row >= 1, col >= 1): The first row, row=1, T(1,col) = col = A000027. When row > col, T(row,col) = row, otherwise (when 1 < row <= col), T(row,col) = row-1.

1, 2, 2, 3, 1, 3, 4, 1, 3, 4, 5, 1, 2, 4, 5, 6, 1, 2, 4, 5, 6, 7, 1, 2, 3, 5, 6, 7, 8, 1, 2, 3, 5, 6, 7, 8, 9, 1, 2, 3, 4, 6, 7, 8, 9, 10, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13

Offset: 1

Antti Karttunen, Feb 10 2014

This is transpose of A237447, please see comments there.

			The top left 9 X 9 corner of this infinite square array:
  1 2 3 4 5 6 7 8 9
  2 1 1 1 1 1 1 1 1
  3 3 2 2 2 2 2 2 2
  4 4 4 3 3 3 3 3 3
  5 5 5 5 4 4 4 4 4
  6 6 6 6 6 5 5 5 5
  7 7 7 7 7 7 6 6 6
  8 8 8 8 8 8 8 7 7
  9 9 9 9 9 9 9 9 8

Antti Karttunen, Table of first 144 antidiagonals of array, flattened

Transpose: A237447.
The leftmost column and the topmost row: A000027. Second row: A054977. Central diagonal: A028310 (note the different starting offsets).
Antidiagonal sums: A074148.
Cf. A237265, A000330, A002024, A002260, A004736, A000124, A010054.

(define (A237448 n) (cond ((= 1 (A010054 (- n 1))) (A002024 n)) ((< (A004736 n) (A002260 n)) (A002260 n)) (else (- (A002260 n) 1))))

As a one-dimensional sequence:
If A010054(n-1) = 1 [that is, if n is in A000124], then a(n) = A002024(n), otherwise, if A004736(n) < A002260(n), a(n) = A002260(n), and if A004736(n) >= A002260(n), a(n) = A002260(n)-1.
Equivalently, as a square array T:
When col < row, T(row,col) = row, for 1 < row <= col, T(row,col) = row-1, and for the first row T(1,col) = col = A000027(col).
Can be computed also as a transposed version of the infinite limit of the finite square arrays in sequence A237265: T(row,col) = A237265((A000330(max(row,col)-1)+1) + (max(row,col)*(col-1)) + (row-1)).

A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

Original entry on oeis.org

0, 1, 2, 1, 2, 0, 1, 3, 3, 2, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 2, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 2, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Paul Curtz, Mar 11 2015

Consider the difference table of a sequence with A000004(n)=0's as main diagonal. (Example: A000045(n).) We call this sequence an autosequence of the first kind.
Based on Pascal's triangle, a(n) =
0,                   T1
1, 2,
1, 2, 0,
1, 3, 3, 2,
etc.
transforms every sequence s(n) in an autosequence of the first kind via the multiplication by the triangle
s0,                  T2
s0, s1,
s0, s1, s2,
s0, s1, s2, s3,
etc.
Examples.
1) s(n) = A198631(n)/A006519(n+1), the second fractional Euler numbers (see A209308). This yields 0*1, 1*1+2*1/2=2, 1*1+2*1/2+0*0=2, 1*1+3*1/2++3*0+2*(-1/4)=2, ... .
The autosequence is 0 followed by 2's or 2*(0,1,1,1,1,1,1,1,... = b(n)).
b(n), the basic autosequence of the first kind, is not in the OEIS (see A140575 and A054977).
2) s(n) = A164555(n)/A027642(n), the second Bernoulli numbers, yields 0,2,2,3,4,5,6,7,... = A254667(n).
Row sums of T1: A062510(n) = 3*A001045(n).
Antidiagonal sums of T1: A111573(n).
With 0's instead of the spaces, every column, i.e.,
0, 1, 1, 1, 1,  1,  1,  1,  1,  1,   1, ...
0, 2, 2, 3, 4,  5,  6,  7,  8,  9,  10, ... = A001477(n) with 0 instead of 1 = A254667(n)
0, 0, 0, 3, 6, 10, 15, 21, 28, 36,  45, ... = A161680(n) with 0 instead of 1
0, 0, 0, 2, 4, 10, 20, 35, 56, 84, 120, ...
etc., is an autosequence of the first kind.
With T(0,0) = 1, it is (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 24 2015

			Triangle starts:
0;
1, 2;
1, 2, 0;
1, 3, 3, 2;
1, 4, 6, 4, 0;
1, 5, 10, 10, 5, 2;
1, 6, 15, 20, 15, 6, 0;
...

Cf. A001045, A001477, A010673, A011973, A054977, A062510, A007318, A197870, A006519, A198631, A140575, A209308, A164555, A027642, A111573, A161680, A254667.

a[n_, k_] := If[k == n, 2*Mod[n, 2], Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 23 2015 *)

a(n) = Pascal's triangle A007318(n) with main diagonal A010673(n) (= period 2: repeat 0, 2) instead of 1's=A000012(n).
a(n) = reversal abs(A140575(n)).
a(n) = A007318(n) + A197870(n+1).
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k>n or if k<0 . - Philippe Deléham, May 24 2015
G.f.: (-1-2*x*y+x^2*y+x^2*y^2)/((x*y+1)*(x*y+x-1)) - 1. - R. J. Mathar, Aug 12 2015

A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 4, 0, 0, 1, 1, 4, 3, 1, 0, 1, 1, 6, 0, 0, 0, 0, 1, 1, 6, 2, 1, 1, 0, 0, 1, 1, 6, 2, 2, 0, 0, 0, 0, 1, 1, 8, 4, 0, 1, 1, 0, 0, 0, 1, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 4, 4, 2, 1, 1, 0, 0, 0, 0, 1, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Apr 17 2007

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 1, 1, 1;
  4, 0, 0, 1, 1;
  4, 3, 1, 0, 1, 1;
  6, 0, 0, 0, 0, 1, 1;
  6, 2, 1, 1, 0, 0, 1, 1;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000010 (alternating row sums), A053158 (row sums).
Cf. A000038, A019590, A054523, A054977.
Cf. A063524, A097806, A126246, A183918.

A054523:= func< n,k | n eq 1 select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
A129479:= func< n,k | k le n-1 select A054523(n,k) + A054523(n,k+1) else 1 >;
[A129479(n,k): k in [1..n], n in [1..16]]; // G. C. Greubel, Feb 11 2024

A054523[n_, k_]:= If[n==1, 1, If[Divisible[n,k], EulerPhi[n/k], 0]];
T[n_, k_]:= If[kA054523[n, j+k], {j,0,1}], 1];
Table[T[n,k],{n,16},{k,n}]//Flatten (* G. C. Greubel, Feb 11 2024 *)

def A054523(n,k):
    if (k==n): return 1
    elif (n%k): return 0
    else: return euler_phi(n//k)
def A129479(n, k):
    if k<0 or k>n: return 0
    elif k==n: return 1
    else: return A054523(n,k) + A054523(n,k+1)
flatten([[A129479(n, k) for k in range(1,n+1)] for n in range(1,17)]) # G. C. Greubel, Feb 11 2024

Sum_{k=1..n} T(n, k) = A053158(n) (row sums).
T(n, 1) = A126246(n).
From G. C. Greubel, Feb 11 2024: (Start)
T(n, k) = A054523(n, k) + A054523(n, k+1) for k < n, otherwise 1.
T(2*n-1, n) = A019590(n).
T(2*n, n) = A054977(n).
T(2*n+1, n) = A000038(n).
T(3*n, n) = A063524(n-1).
T(3*n-2, n) = A183918(n+2).
Sum_{k=1..n} (-1)^(k-1) * T(n, k) = A000010(n). (End)

A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 4, 1, 0, 8, 12, 6, 1, 0, 16, 32, 24, 8, 1, 0, 32, 80, 80, 40, 10, 1, 0, 64, 192, 240, 160, 60, 12, 1, 0, 128, 448, 672, 560, 280, 84, 14, 1, 0, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 0, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 0

Offset: 0

Philippe Deléham, Dec 14 2011

A071919*A007318 as infinite lower triangular matrices.
A129186*A038207 as infinite lower triangular matrices.
From Paul Curtz, Nov 12 2019: (Start)
If a new main diagonal of 0's is added to the triangle, then for this variant the following propositions hold:
   The first column is A166444.
   The second column is A139756.
   The antidiagonal sums are A000129 (Pell numbers).
   The row sums are (-1)^n*A141413.
   The signed row sums are 0 followed by 1's, autosequence companion to A054977.
(End)

			Triangle begins:
   1;
   1,  0;
   2,  1,  0;
   4,  4,  1,  0;
   8, 12,  6,  1,  0;
  16, 32, 24,  8,  1, 0;
  32, 80, 80, 40, 10, 1, 0;
  ...

Essentially the same as A038207, A062715, A065109.
Cf. A001787, A001788, A139756, A000129 (antidiagonals sums).
Cf. A141413, A166444, A054977.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n<2, 1-k, 2*T(n-1, k) +T(n-1, k-1)))
    end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Nov 08 2019

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2 T[n-1, k] + T[n-1, k-1];
T[0, 0] = T[1, 0] = 1; T[1, 1] = 0; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0)=T(1,0)=1 and T(1,1)=0 .
G.f.: (1-(1+y)*x)/(1-(2+y)*x).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A019590(n+1), A000012(n), A011782(n), A133494(n) for x = -2, -1, 0, 1 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A133494(n), A020699(n) for x = 0, 1, 2 respectively.
T(2n,n) = A069720(n).

A256096 Expansion of (4+3x)/(1+3x).

Original entry on oeis.org

4, -9, 27, -81, 243, -729, 2187, -6561, 19683, -59049, 177147, -531441, 1594323, -4782969, 14348907, -43046721, 129140163, -387420489, 1162261467, -3486784401, 10460353203, -31381059609, 94143178827, -282429536481, 847288609443, -2541865828329, 7625597484987

Offset: 0

Wolfdieter Lang, Mar 23 2015

This is the Z-sequence of the Riordan triangle ((1+2*x)/(1-x)^2, x/(1-x)). See A135857.

The expansion can be seen as a special case of the family of generating functions 1+1/(x+1/k) for k>=1 which relates this sequence to A054977 and A198633 (neglecting questions of sign). - Peter Luschny, Mar 24 2015

Winston de Greef, Table of n, a(n) for n = 0..2080
Index entries for linear recurrences with constant coefficients, signature (-3).

Cf. A135857, A054977, A198633.

Join[{4},NestList[-3#&,-9,30]] (* Harvey P. Dale, Jan 27 2023 *)

a(n)=if(!n, 4, (-1)^n*3^(n+1)) \\ Winston de Greef, Mar 19 2023

O.g.f.: (4+3*x)/(1+3*x).
a(0) = 4; for n >= 1, a(n) = (-1)^n*3^(n+1).
a(0) = 4, a(1) = -9; for n >= 2, a(n) = -3*a(n-1).
E.g.f.: 1 + 3*exp(-3*x). - Alejandro J. Becerra Jr., Jan 28 2021

A277627 Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 0, 3, 5, 1, 0, 0, 0, 0, 0, 0, 0, 6, 6, 1, 0, 0, 0, 0, 0, 0, 0, 1, 10, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 15, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 21, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 28, 10, 1

Offset: 0

Paul Curtz, Oct 24 2016

In other words, for n > 0 the column k lists 2*k+1 zeros together with the partial sums of the positive terms of column k-1. - Omar E. Pol, Oct 25 2016
Comments from the author:
1) ZSPEC =
  0, 0, 0, 0, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, 0, 0, 0, ...
  1, 2, 0, 0, 0, 0, 0, 0, ...
  1, 3, 1, 0, 0, 0, 0, 0, ...
  1, 4, 3, 0, 0, 0, 0, 0, ...
  1, 5, 6, 1, 0, 0, 0, 0, ...
etc.
The columns are the autosequences of the first kind of the title (column 1: 0, 0, followed by A001477(n); column 2: 0, 0, 0, 0, followed by A000217(n), etc) .
The positive terms are the Pascal triangle written by diagonals (A011973).
First column: A060576(n+1). Or A057427(n), n>-1, thanks to Omar E. Pol.
Row sums: A000045(n), autosequence of the first kind.
Alternated row sums and subtractions: 0, 1, 1, 0, -1, -1, 0 = A128834(n), autosequence of the first kind.
Antidiagonal sums: 0, 1, 1, 1, 2, 3, 4, 6, ... = A078012(n+2).
Application.
Numbers in triangle leading to the Genocchi numbers -A226158(n).
We multiply the columns of ZSPEC by d(n) = 1, -1, 2, -8, 56, -608, ... from A005439.
Hence, with only the first 0,
  0,
  1,
  1,
  1, -1,
  1, -2,
  1, -3,  2,
  1, -4,  6,
  1, -5, 12,   -8,
  1, -6, 20,  -32,
  1, -7, 30,  -80,  56,
  1, -8, 42, -160, 280,
  etc.
The row sums is -A226158(n).
2) Now consider the case of the autosequences of the second kind.
First step.
            2, 1, 1,  1,  1,  1, ...        = A054977(n)
      0, 0, 2, 3, 4,  5,  6,  7, ...        = A199969(n) with offset 0
0, 0, 0, 0, 2, 5, 9, 14, 20, 27, ...        see A000096
etc.
The positive terms are ASPEC in A191302. By triangle, they are either A029653(n) with A029653(0) = 2 instead of 1 or A029635(n).
Second step. YSPEC =
  2, 0,  0, 0, 0, 0, ...
  1, 0,  0, 0, 0, 0, ...
  1, 2,  0, 0, 0, 0, ...
  1, 3,  0, 0, 0, 0, ...
  1, 4,  2, 0, 0, 0, ...
  1, 5,  5, 0, 0, 0, ...
  1, 6,  9, 2, 0, 0, ...
  1, 7, 14, 7, 0, 0, ...
  etc.
Diagonals by triangle: A029635(n).
This is the companion to ZSPEC.
Row sums: A000032(n), autosequence of the second kind.
Alternated row sums and subtractions: period 6 repeat 2, 1, -1, -2, -1, 1 = A087204(n), autosequence of the second kind.
Application.
Numbers in triangle leading to A230324(n), a companion to -A226158(n).
We multiply the columns of YSPEC by d(n) 1, -1, 2, -8, 56, ... (see above).
Hence, without zeros:
  2,
  1,
  1, -2,
  1, -3,
  1, -4,  4,
  1, -5, 10,
  1, -6, 18,  -16,
  1, -7, 28,  -56,
  1, -8, 40, -128, 112,
  1, -9, 54, -240, 504,
  etc.
The row sum is A230324(n).

G. C. Greubel, Table of n, a(n) for the first 25 antidiagonals, flattened
OEIS Wiki, Autosequence

Cf. A011973 (without 0's), A007318 (Pascal's triangle).
Cf. A000045 (row sums), A078012 (antidiagonal sums).
Columns: A060576 or A057427 (k=0), A001477 (k=1), A000217 (k=2).
Cf. A000004, A000012, A000027, A000032, A005439, A029635, A029653, A054977, A087204, A128834, A191302, A199969, A226158, A230324.

kMax = 13; col[0] = Join[{0}, Array[1&, kMax]]; col[k_] := col[k] = Join[{0, 0}, col[k-1][[1 ;; -3]] // Accumulate]; T[n_, k_] := col[k][[n+1]]; Table[T[n-k, k], {n, 0, kMax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 15 2016 *)

Better definition from Omar E. Pol, Oct 25 2016

A328284 An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485

Offset: 0

Paul Curtz, Oct 11 2019

nonn

OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000129, A001045, A054977, A139756, A141413, A166444, A183190.

Cf. A000079, A005578, A063524, A086445, A139818, A257113.

a[n_] := If[n>3, (2^(n-3) + (-1)^n)/3, If[n == 2, 1, 0]]; (* Jean-François Alcover, Oct 16 2019 *)

a(n) is the fourth row of the following array:
  0,  0,  0,  0, 0, 1, 3, 7, 14, 27, 51, 97, ...
  0,  0,  0,  0, 1, 2, 4, 7, 13, 24, 46, 89, ... = A086445
  0,  0,  0,  1, 1, 2, 3, 6, 11, 22, 43, 86, ... = 0, 0, 0, A005578(n)
  0,  0,  1,  0, 1, 1, 3, 5, 11, 21, 43, 85, ... = a(n)
  0,  1, -1,  1, 0, 2, 2, 6, 10, 22, 42, 86, ...
  1, -2,  2, -1, 2, 0, 4, 4, 12, 20, 44, 84, ...
From the main diagonal onward, every row is an autosequence of the first kind.
From Stefano Spezia, Oct 16 2019: (Start)
O.g.f.: x^2*(-1 + x + x^2)/(-1 + x + 2*x^2).
E.g.f.: (1/24)*exp(-x)*(8 - 9*exp(x) + exp(3*x) + 6*exp(x)*x + 6*exp(x)*x^2).
a(n) = a(n-1) + 2*a(n-2) for n > 4. (End)
a(n) = Sum_{k=0..n-1} A183190(n-k-2, n-2*k-2). - Jean-François Alcover, Nov 10 2019

Partially edited by Peter Luschny, Nov 12 2019

A375038 Irregular triangle read by rows T(n,k), n >= 2, k >= 1, in which row n lists the nonmiddle divisors of n.

Original entry on oeis.org

2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 2, 6, 12, 1, 13, 1, 2, 7, 14, 1, 15, 1, 2, 8, 16, 1, 17, 1, 2, 6, 9, 18, 1, 19, 1, 2, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 8, 12, 24, 1, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 14, 28

Offset: 2

Omar E. Pol, Jul 28 2024

Except the 1, all positive integers have nonmiddle divisors.

The nonmiddle divisors of n are here the divisors of n that are not in the half-open interval [sqrt(n/2), sqrt(n*2)).

			Triangle begins starting in row n = 2:
  2;
  1, 3;
  1, 4;
  1, 5;
  1, 6;
  1, 7;
  1, 4, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 6, 12;
  ...
For n = 12 the divisors of 12 are [1, 2, 3, 4, 6, 12] and the middle divisors are [3, 4], so the nonmiddle divisors are [1, 2, 6, 12], the same as the row n = 12 of the triangle.

Amiram Eldar, Table of n, a(n) for n = 2..6375 (rows for n = 2..1000, flattened)

Nonzero terms of A375037.
The sum of row n is A302433(n).
The number of terms in row n is A067743(n).
Column 1 gives A054977.
Cf. A027750, A067742, A071090, A299761, A303297.

row[n_] := Select[Divisors[n], !(Sqrt[n/2] <= # < Sqrt[2*n]) &]; Table[row[n], {n, 2, 28}] // Flatten (* Amiram Eldar, Jul 29 2024 *)

A174419 Numerators T(0,k) of a top row sequence which generates a signed variant (-1)^n*T(n,0) of itself in the column k=0 under repeated application of the Akiyama-Tanigawa transform.

Original entry on oeis.org

0, 1, 3, 29, 213, 36361, 5004267, 161159569259, 1604875494550299, 700591444676447407855, 272366765005761133289834097, 441056613421971051554626329901900903, 48264034659082736983682770426524745021503, 162486296853709899698219310156295323853814636455303

Offset: 0

Paul Curtz, Mar 19 2010

The sequence contains the numerators of the top row in the following table, where successive rows are constructed by iteration of the Akiyama-Tanigawa transform:
0, 1, 3, 29/5, 213/23,...
-1, -4, -42/5, -1592/115, -55070/2737,..
3, 44/5, 1878/115, 343608/13685, 68612650/1967903,..
-29/5, -1732/115, -360378/13685, -22590376/578795, -74842810298/1416609031,...
213/23, 61708/2737, 74954766/1967903, 2737355924568/49581316085,...
The associated denominators in the first row are 1, 1, 1, 5, 23, 2737, 281129, 7083045155,...
The top row is designed to reproduce itself (up to alternating sign) in the leftmost column under the transformation.
There are other examples of sequences quasi-preserved under the Akiyama-Tanigawa transform: if the first row were T(0,k)= A054977(k), the first column would be identical to the first row (no sign flips in this example).
Another (trivial) example is the all-0 sequence, which produces a table containing only zeros.

D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

nmax := 10 ;
T := array(0..nmax,0..nmax) ;
T[0,0] := 0 ; T[0,1] := 1 ; T[1,0] := -1 ;
for n from 2 to nmax do
        T[0,n] := x ;
        for r from 1 to n do k := n-r ; T[r,k] := (k+1)*(T[r-1,k]-T[r-1,k+1]) ;
        end do:
        y := solve( T[n,0] = (-1)^n*T[0,n]) ; T[0,n] :=  y;
        for r from 1 to n do k := n-r ; T[r,k] := (k+1)*(T[r-1,k]-T[r-1,k+1]) ;
        end do:
end do:
seq( numer(T[0,i]),i=0..nmax) ; # R. J. Mathar, Dec 02 2010

nmax=10; t[0,0]=0; t[0,1]=1; t[1,0]=-1; For[n=2, n<= nmax, n++, t[0,n]=x; For[r=1, r<=n, r++, k=n-r; t[r,k]=(k+1)*(t[r-1,k]-t[r-1,k+1]);]; y=x/.Solve[t[n,0]==(-1)^n*t[0,n]]//First; t[0,n]=y; For[r=1, r<=n, r++, k=n-r; t[r,k]=(k+1)*(t[r-1,k]-t[r-1,k+1]);]]; Table[ t[0,i],{i,0,nmax}] // Numerator (* Jean-François Alcover, Sep 18 2012, translated from Maple *)

A277078 Triangular array similar to A255935 but with 0's and 2's swapped in the trailing diagonal. The columns alternate in signs.

Original entry on oeis.org

2, 1, 0, 1, -2, 2, 1, -3, 3, 0, 1, -4, 6, -4, 2, 1, -5, 10, -10, 5, 0, 1, -6, 15, -20, 15, -6, 2, 1, -7, 21, -35, 35, -21, 7, 0, 1, -8, 28, -56, 70, -56, 28, -8, 2, 1, -9, 36, -84, 126, -126, 84, -36, 9, 0, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 2

Offset: 0

Paul Curtz, Oct 23 2016

a(n)=
2,
1,  0,
1, -2, 2,
1, -3, 3,  0,
1, -4, 6, -4, 2,
etc.
transforms every sequence s(n) in an autosequence of the second kind via the multiplication by the triangle
s0,           T2
s0, s1,
s0, s1, s2,
s0, s1, s2, s3,
etc.
which is the reluctant form of s(n).
Example.
s(n) = A131577(n) = 0, 1, 2, 4, ... .
The multiplication gives 0, 0, 2, 3, 8, 15, 32, 63, ... = 0 followed by A166920.
a(n) comes from alternate sum and difference of s(n) and t(n), its inverse binomial transform. In the example (t(n) = periodic 2: repeat 0, 1) the first terms are: 0+0, 1-1, 2+0, 4-1, 8+0, 16-1, 32+0, 64-1, ... .

Cf. A000035, A007318, A054977, A131577, A166920, A197870, A255935.

a[n_, k_] := If[k == n, 2 - 2*Mod[n, 2], (-1)^k*Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 16 2016 *)

a(n) = A007318(n) - A197870(n+1).

cp's OEIS Frontend

A237448 Square array T(row >= 1, col >= 1): The first row, row=1, T(1,col) = col = A000027. When row > col, T(row,col) = row, otherwise (when 1 < row <= col), T(row,col) = row-1.

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A255935 Triangle read by rows: a(n) = Pascal's triangle A007318(n) + A197870(n+1).

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A129479 Triangle read by rows: A054523 * A097806 as infinite lower triangular matrices.

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A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A256096 Expansion of (4+3*x)/(1+3*x).

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A277627 Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.

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A328284 An extension of the Jacobsthal numbers: 0, 0, 1, followed by A001045.

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A375038 Irregular triangle read by rows T(n,k), n >= 2, k >= 1, in which row n lists the nonmiddle divisors of n.

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A256096 Expansion of (4+3x)/(1+3x).