A000012 -id:A000012 - cp's OEIS frontend

A131060 3A007318 - 2A000012 as infinite lower triangular matrices.

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 16, 10, 1, 1, 13, 28, 28, 13, 1, 1, 16, 43, 58, 43, 16, 1, 1, 19, 61, 103, 103, 61, 19, 1, 1, 22, 82, 166, 208, 166, 82, 22, 1, 1, 25, 106, 250, 376, 376, 250, 106, 25, 1, 1, 28, 133, 358, 628, 754, 628, 358, 133, 28, 1

Offset: 0

Gary W. Adamson, Jun 13 2007

Row sums = A097813: (1, 2, 6, 16, 38, 84, 178, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  4,  1;
  1,  7,  7,  1;
  1, 10, 16, 10,  1;
  1, 13, 28, 28, 13,  1;
  1, 16, 43, 58, 43, 16,  1;
  ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A109128, A123203, A131061, A131063, A131064, A131065, A131066, A131067, A131068.

[3*Binomial(n,k) -2: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020

A131060:= (n,k) -> 3*binomial(n, k)-2; seq(seq(A131060(n, k), k = 0..n), n = 0.. 10); # G. C. Greubel, Mar 12 2020

T[n_, k_] = 3*Binomial[n, k] -2; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 20 2008 *)

[[3*binomial(n,k) -2 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

T(n,k) = 3*binomial(n,k) - 2. - Roger L. Bagula, Aug 20 2008

More terms from Roger L. Bagula, Aug 20 2008

A131818 A130296 + A002260 - A000012. Triangle read by rows: row n consists of n, 2, 3, 4, ..., n.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jul 18 2007

Row sums = A034856; (1, 4, 8, 13, 19, 26, 34, ...).

			First few rows of the triangle:
  1;
  2, 2;
  3, 2, 3;
  4, 2, 3, 4;
  5, 2, 3, 4, 5;
  6, 2, 3, 4, 5, 6;
  7, 2, 3, 4, 5, 6, 7;
  ...

Cf. A130296, A002260, A034856.

Table[Join[{n},Range[2,n]],{n,15}]//Flatten (* Harvey P. Dale, Feb 24 2021 *)

t(n, k) = if (k==1, n, k); \\ Michel Marcus, Feb 12 2014

from math import isqrt, comb
def A131818(n):
    y = (m:=isqrt(k:=n-1<<1))+(k>m*(m+1))
    return n-comb(y,2) # Chai Wah Wu, Jul 07 2025

A130296 + A002260 - A000012 as infinite lower triangular matrices.

T(n, 1) = n, T(n, k) = k for k > 1. - Michel Marcus, Feb 12 2014

More terms from Michel Marcus, Feb 12 2014

A131110 A000012 * A133084.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 10, 6, 5, 1, 15, 10, 15, 5, 1, 21, 15, 35, 15, 7, 1, 28, 21, 70, 35, 28, 7, 1, 36, 28, 126, 70, 84, 28, 9, 1, 45, 36, 210, 126, 210, 84, 45, 9, 1, 55, 45, 330, 210, 462, 210, 165, 45, 11, 1

Offset: 1

Gary W. Adamson, Sep 08 2007

Row sums give A033484.

Duplicate of A133093. - Georg Fischer, Oct 10 2021

			First few rows of the triangle are:
1;
3, 1;
6, 3, 1;
10, 6, 5, 1;
15, 10, 15, 5, 1;
21, 15, 35, 15, 7, 1;
28, 21, 70, 35, 28, 7, 1;
...

Cf. A133084, A033484.

T4(n, k) = if(k == n, 1, (1  - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)); \\ A133084
N=10; matrix(N, N, n, k, if(n>=k, 1))*matrix(N, N, n, k, T4(n,k)) \\ Michel Marcus, Oct 11 2021

A000012 * A133084 as infinite lower triangular matrices.

a(46) corrected by Georg Fischer, Oct 10 2021

A137650 Triangle read by rows, A008277 * A000012.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 15, 14, 7, 1, 52, 51, 36, 11, 1, 203, 202, 171, 81, 16, 1, 877, 876, 813, 512, 162, 22, 1, 4140, 4139, 4012, 3046, 1345, 295, 29, 1, 21147, 21146, 20891, 17866, 10096, 3145, 499, 37, 1, 115975, 115974, 115463

Offset: 1

Gary W. Adamson, Feb 01 2008

Left column = Bell numbers (A000110) starting (1, 2, 5, 15, 52, 203, ...). Row sums = A005493(n+1): (1, 3, 10, 37, 151, 674, ...).

Corresponding to the generalized Stirling number triangle of first kind A049444. - Peter Luschny, Sep 18 2011

			First few rows of the triangle are
    1;
    2,   1;
    5,   4,   1;
   15,  14,   7,   1;
   52,  51,  36,  11,   1;
  203, 202, 171,  81,  16,   1;
  877, 876, 813, 512, 162,  22,   1;
  ...

Cf. A000110, A008277, A005493, A049444.

A similar triangle is A133611.

A137650_row := proc(n) local k,i;
add(add(combinat[stirling2](n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,k),k=0..n-1) end:
seq(print(A137650_row(n)),n=1..7); # Peter Luschny, Sep 18 2011

row[n_] := Table[StirlingS2[n, k], {k, 0, n}] // Reverse // Accumulate // Reverse // Rest;
Array[row, 10] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

A008277 * A000012 as infinite lower triangular matrices. Partial sums of A008277 rows starting from the right.

A128379 A000012^23 * A000594.

Original entry on oeis.org

1, -1, -24, 0, 276, 300, -1748, -4300, 4278, 29026, 22724, -94668, -242398, -18722, 856980, 1472252, -384491, -5299269, -7824968, 2088032, 25655442, 38814478, -69160, -99735912, -175711283, -68736397, 294769680, 686373176, 562588924, -513324396, -2155273788, -2808874356

Offset: 1

Gary W. Adamson, Feb 28 2007

sign

Conjecture: given A000012^k * A000594, k=23 and 24 are the only k's generating sequences with zeros. k = 24 in A128378: (1, 0, -24, -24, 252, 552, -1196, -5496, ...).

Cf. A000594, A128378, A144248, A144249.

Nest[Accumulate, RamanujanTau[Range[32]], 23] (* Amiram Eldar, Jan 08 2025 *)

A000012 (partial sum operator) performed 23 times on A000594.

A131816 Triangle read by rows: A130321 + A059268 - A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...), A059268 = (1; 1,2; 1,2,4; ...) and A000012 = (1; 1,1; 1,1,1; ...).

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 8, 5, 5, 8, 16, 9, 7, 9, 16, 32, 17, 11, 11, 17, 32, 64, 33, 19, 15, 19, 33, 64, 128, 65, 35, 23, 23, 35, 65, 128, 256, 129, 67, 39, 31, 39, 67, 129, 256, 512, 257, 131, 71, 47, 47, 71, 131, 257, 512, 1024, 513, 259, 135, 79, 63, 79, 135, 259, 513, 1024

Offset: 0

Gary W. Adamson, Jul 18 2007

Row sums = A000295: (1, 4, 11, 26, 57, 120, ...).
If we regard the sequence as an infinite square array read by diagonals then it has the formula U(n,k) = (2^n + 2^k)/2 - 1. This appears to coincide with the number of n X k 0..1 arrays colored with only straight tiles, and new values 0..1 introduced in row major order, i.e., no equal adjacent values form a corner. (Fill the array with 0's and 1's. There must never be 3 adjacent identical values making a corner, only same values in a straight line.) Some solutions with n = k = 4 are:
  0 1 0 1     0 1 0 0     0 0 0 1     0 0 1 1     0 1 0 1
  1 0 1 0     1 0 1 1     1 1 1 0     1 1 0 0     1 0 1 0
  1 0 1 0     0 1 0 0     0 0 0 1     0 0 1 1     0 1 0 1
  0 1 0 1     1 0 1 1     1 1 1 0     1 1 0 0     0 1 0 1
(Observation from R. H. Hardin, cf. link.) - M. F. Hasler and N. J. A. Sloane, Feb 26 2013

			First few rows of the triangle:
    1;
    2,  2;
    4,  3,  4;
    8,  5,  5,  8;
   16,  9,  7,  9, 16;
   32, 17, 11, 11, 17, 32;
   64, 33, 19, 15, 19, 33, 64;
  128, 65, 35, 23, 23, 35, 65, 128;
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
R. H. Hardin, Post to the SeqFan list, Feb 26 2013

Cf. A130321, A059268, A000012.

Row sums give A000295(n+2).

a131816 n k = a131816_tabl !! n !! k
a131816_row n = a131816_tabl !! n
a131816_tabl = map (map (subtract 1)) $
   zipWith (zipWith (+)) a130321_tabl a059268_tabl
-- Reinhard Zumkeller, Feb 27 2013

Table[Table[((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2, {m, 0, n}], {n, 0, 10}]; Flatten[%] (* Roger L. Bagula, Oct 16 2008 *)

T(n,m) = ((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2. - Roger L. Bagula, Oct 16 2008

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A134541 Triangle read by rows: A000012 * A054525 regarded as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, -1, 0, 1, 1, -2, 0, 1, 1, 1, -1, -1, 0, 1, 1, 1, -2, -1, 0, 1, 1, 1, 1, -2, -1, 0, 0, 1, 1, 1, 1, -2, -1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -1, 0, 0, 1, 1, 1, 1, 1, -2, -2, -1, 0, 0, 1, 1, 1, 1, 1, 1, -2, -1, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Oct 31 2007

Row sums = 1.
Left border = A002321, the Mertens function.
A134541 * [1,2,3,...] = A002088: (1, 2, 4, 6, 10, 12, 18, 22, ...).

			First few rows of the triangle:
   1;
   0,  1;
  -1,  1,  1;
  -1,  0,  1, 1;
  -2,  0,  1, 1, 1;
  -1, -1,  0, 1, 1, 1;
  -2, -1,  0, 1, 1, 1, 1;
  -2, -1,  0, 0, 1, 1, 1, 1;
  -2, -1, -1, 0, 1, 1, 1, 1, 1;
  -1, -2, -1, 0, 0, 1, 1, 1, 1, 1;
  ...

Cf. A000012, A054525, A002321, A002088, A134541.

Matrix inverse of A176702. - Mats Granvik, Apr 24 2010

Clear[t, s, n, k, z, x]; z = 1; nn = 10; t[n_, k_] := t[n, k] = If[n >= k, If[k == 1, 1 - Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}], t[Floor[n/k], 1]], 0]; Flatten[Table[Table[Limit[t[n, k], s -> z], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Jul 22 2012 *) (* updated Mats Granvik, Apr 10 2016 *)

Recurrence: T(n, k) = If n >= k then If k = 1 then 1 - Sum_{i=1..n-1} T(n, k + i)/(i + 1)^(s - 1) else T(floor(n/k) else 1)) else 0). - Mats Granvik, Apr 17 2016

More terms from Amiram Eldar, Jun 09 2024

A137948 Triangle read by rows, A000012 * A136579.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 4, 6, 5, 4, 6, 12, 24, 6, 5, 8, 18, 48, 120, 7, 6, 10, 24, 72, 240, 720, 8, 7, 12, 30, 96, 360, 1440, 5040, 9, 8, 14, 36, 120, 480, 2160, 10080, 40320, 10, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880

Offset: 0

Gary W. Adamson, Feb 28 2008

Row sums = A014144 starting (1, 3, 7, 17, 51, 205, ...).

T(n,k) = A245334(n,k) / A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Aug 31 2014

			First few rows of the triangle:
  1;
  2, 1;
  3, 2,  2;
  4, 3,  4,  6;
  5, 4,  6, 12, 24;
  6, 5,  8, 18, 48, 120;
  7, 6, 10, 24, 72, 240, 720;
  ...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A136579, A014144.

Cf. A245334, A007318.

a137948 n k = a137948_tabl !! n !! k
a137948_row n = a137948_tabl !! n
a137948_tabl = zipWith (zipWith div) a245334_tabl a007318_tabl
-- Reinhard Zumkeller, Aug 31 2014

As infinite lower triangular matrices, A000012 * A136579, where A000012 = (1; 1,1; 1,1,1; ...) and A136579 = (1; 1,1; 1,1,2; 1,1,2,6; 1,1,2,6,24; ...).

T(n,k) = (n+1-k) * k! for 0 <= k <= n. - Werner Schulte, Oct 06 2020

Offset changed by Reinhard Zumkeller, Aug 31 2014

A127096 Triangle T(n,m) = A000012*A127094 read by rows.

Original entry on oeis.org

1, 3, 1, 6, 1, 1, 10, 1, 3, 1, 15, 1, 3, 1, 1, 21, 1, 3, 4, 3, 1, 28, 1, 3, 4, 3, 1, 1, 36, 1, 3, 4, 7, 1, 3, 1, 45, 1, 3, 4, 7, 1, 6, 1, 1, 55, 1, 3, 4, 7, 6, 6, 1, 3, 1, 66, 1, 3, 4, 7, 6, 6, 1, 3, 1, 1, 78, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 91, 1, 3, 4, 7, 6, 12, 1, 7, 4, 3, 1, 1, 105, 1, 3, 4, 7, 6, 12, 8, 7, 4, 3, 1, 3, 1

Offset: 1

Gary W. Adamson, Jan 05 2007

Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127094 from the right.

			First few rows of the triangle are:
   1;
   3, 1,
   6, 1, 1;
  10, 1, 3, 1;
  15, 1, 3, 1, 1;
  21, 1, 3, 4, 3, 1;
  28, 1, 3, 4, 3, 1, 1;
  ...

Cf. A127093, A127094, A123229, A024916 (row sums), A000203, A126988.

A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
A127094 := proc(n,m) A127093(n, n-m+1) ; end:
A127096 := proc(n,m) add( A127094(j,m),j=m..n) ; end:
for n from 1 to 15 do for m from 1 to n do printf("%d,",A127096(n,m)) ; od: od: # R. J. Mathar, Aug 18 2009

T[n_, m_] := Sum[1 + Mod[j, m - j - 1] - Mod[1 + j, m - j - 1], {j, m, n}];
Table[T[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 15 2023 *)

T(n,m) = Sum_{j=m..n} A000012(n,j)*A127094(j,m) = Sum_{j=m..n} A127094(j,m).

Edited and extended by R. J. Mathar, Aug 18 2009

A130330 Triangle read by rows, the matrix product A130321 * A000012, both taken as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 7, 3, 1, 15, 7, 3, 1, 31, 15, 7, 3, 1, 63, 31, 15, 7, 3, 1, 127, 63, 31, 15, 7, 3, 1, 255, 127, 63, 31, 15, 7, 3, 1, 511, 255, 127, 63, 31, 15, 7, 3, 1, 1023, 511, 255, 127, 63, 31, 15, 7, 3, 1, 2047, 1023, 511, 255, 127, 63, 31, 15, 7, 3, 1

Offset: 0

Gary W. Adamson, May 24 2007

Row sums are A000295: (1, 4, 11, 26, 57, 120, 247, ...), the Eulerian numbers.
T(n,k) is the number of length n+1 binary words containing at least two 1's such that the first 1 is preceded by exactly (k-1) 0's. T(3,2) = 3 because we have: 0101, 0110, 0111. - Geoffrey Critzer, Dec 31 2013
Call this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A110441. - Peter Bala, Jul 22 2014
From Wolfdieter Lang, Oct 28 2019:(Start)
This triangle gives the solution of the following problem. Iterate the function f(x) = (x - 1)/2 to obtain f^{[k]}(x) = (x - (2^(k+1) - 1))/2^(k+1), for k >= 0. Find the positive integer x values for which the iterations stay integer and reach 1. Only odd integers x qualify, and the answer is x = x(n) = 2*T(n, 0) = 2*(2^(n+1) - 1), with the iterations T(n,0), ..., T(n,n) = 1.
This iteration is motivated by a problem posed by Johann Peter Hebel (1760 - 1826) in "Zweites Rechnungsexempel" from 1804, with the solution x = 31 corresponding to row n = 3 [15 7 3 1]. The egg selling woman started with 31 = T(4, 0) eggs and after four customers obtained, one after the other, always a number of eggs which was one half of the woman's remaining number of eggs plus 1/2 (selling only whole eggs, of course) she had one egg left. See the link and reference. [For Hebel's first problem see a comment in A000225.]
(End)

			First few rows of the triangle T(n, k):
n\k     0    1    2    3   4   5   6  7  8  9 10 11 12 ...
0:      1
1:      3    1
2:      7    3    1
3      15    7    3    1
4:     31   15    7    3   1
5:     63   31   15    7   3   1
6:    127   63   31   15   7   3   1
7:    255  127   63   31  15   7   3  1
8:    511  255  127   63  31  15   7  3  1
9:   1023  511  255  127  63  31  15  7  3  1
10:  2047 1023  511  255 127  63  31 15  7  3  1
11:  4095 2047 1023  511 255 127  63 31 15  7  3  1
12:  8191 4095 2047 1023 511 255 127 63 31 15  7  3  1
... reformatted and extended. - _Wolfdieter Lang_, Oct 28 2019

Johann Peter Hebel, Gesammelte Werke in sechs Bänden, Herausgeber: Jan Knopf, Franz Littmann und Hansgeorg Schmidt-Bergmann unter Mitarbeit von Ester Stern, Wallstein Verlag, 2019. Band 3, S. 36-37, Solution, S. 40-41. See also the link below.

Reinhard Zumkeller, Table of n, a(n) for n = 0..11324 [first 150 rows; offset shifted by _Georg Fischer_, Oct 29 2019]
Johann Peter Hebel, Zweites Rechnungsexempel., 1804; Solution: Auflösung des zweiten Rechnungsexempels. , 1805.

Cf. A130321, A000012, A000225. A110441.

a130330 n k = a130330_row n !! (k-1)
a130330_row n = a130330_tabl !! (n-1)
a130330_tabl = iterate (\xs -> (2 * head xs + 1) : xs) [1]
-- Reinhard Zumkeller, Mar 31 2012

nn=12;a=1/(1- x);b=1/(1-2x);Map[Select[#,#>0&]&,Drop[CoefficientList[Series[a x^2 b/(1-y x),{x,0,nn}],{x,y}],2]]//Grid  (* Geoffrey Critzer, Dec 31 2013 *)

A130321 * A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...) and A000012 = (1; 1,1; 1,1,1; ...).
In every column k with offset n = k: 2^(m+1) - 1 = A000225(m+1) = (1, 3, 7, 15, ...), for m >= 0.
G.f.: 1/((1-y*x)*(1-x)*(1-2x)). - Geoffrey Critzer, Dec 31 2013
T(n, k) = 2^((n - k) + 1) - 1, n >= 0, k = 0..n. - Wolfdieter Lang, Oct 28 2019

More terms from Geoffrey Critzer, Dec 31 2013

Edited by Wolfdieter Lang, Oct 28 2019

cp's OEIS Frontend

A131060 3*A007318 - 2*A000012 as infinite lower triangular matrices.

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A131818 A130296 + A002260 - A000012. Triangle read by rows: row n consists of n, 2, 3, 4, ..., n.

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A131110 A000012 * A133084.

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A137650 Triangle read by rows, A008277 * A000012.

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A128379 A000012^23 * A000594.

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A131816 Triangle read by rows: A130321 + A059268 - A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...), A059268 = (1; 1,2; 1,2,4; ...) and A000012 = (1; 1,1; 1,1,1; ...).

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A134541 Triangle read by rows: A000012 * A054525 regarded as infinite lower triangular matrices.

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A131060 3A007318 - 2A000012 as infinite lower triangular matrices.