A000004 -id:A000004 - cp's OEIS frontend

A195152 Square array read by antidiagonals with T(n,k) = n((k+2)n-k)/2, n=0, +- 1, +- 2,..., k>=0.

0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 4, 5, 3, 1, 0, 9, 7, 6, 4, 1, 0, 9, 12, 10, 7, 5, 1, 0, 16, 15, 15, 13, 8, 6, 1, 0, 16, 22, 21, 18, 16, 9, 7, 1, 0, 25, 26, 28, 27, 21, 19, 10, 8, 1, 0, 25, 35, 36, 34, 33, 24, 22, 11, 9, 1, 0, 36, 40, 45, 46, 40, 39, 27, 25, 12, 10, 1, 0

Offset: 0

Omar E. Pol, Sep 14 2011

Also, column k lists the partial sums of the column k of A195151. The first differences in row n are always the n-th term of the triangular numbers repeated 0,0,1,1,3,3,6,6,... ([0,0] together with A008805).

Also, for k >= 1, this is a table of generalized polygonal numbers since column k lists the generalized m-gonal numbers, where m = k+4, for example: if k = 1 then m = 5, so the column 1 lists the generalized pentagonal numbers A001318 (see example).

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,...
.  4,   5,   6,   7,   8,   9,  10,  11,  12,  13,...
.  4,   7,  10,  13,  16,  19,  22,  25,  28,  31,...
.  9,  12,  15,  18,  21,  24,  27,  30,  33,  36,...
.  9,  15,  21,  27,  33,  39,  45,  51,  57,  63,...
. 16,  22,  28,  34,  40,  46,  52,  58,  64,  70,...
. 16,  26,  36,  46,  56,  66,  76,  86,  96, 106,...
. 25,  35,  45,  55,  65,  75,  85,  95, 105, 115,...
. 25,  40,  55,  70,  85, 100, 115, 130, 145, 160,...
...

Rows: A000004, A000012, A000027.
Column 0 gives A008794, except its first term.
Columns >= 1: A001318, (A000217), A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818.
Cf. A008805, A152151.

T(n,k) = (k+2)*n*(n+1)/8+(k-2)*((2*n+1)*(-1)^n-1)/16, n >= 0 and k >= 0. - Omar E. Pol, Oct 01 2011

A212208 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square diagonal grid graph DG_(n,n), highest powers first.

Original entry on oeis.org

1, 0, 1, -6, 11, -6, 0, 1, -20, 174, -859, 2627, -5082, 6048, -4023, 1134, 0, 1, -42, 825, -10054, 85011, -528254, 2491825, -9084089, 25795983, -57031153, 97292827, -125639547, 118705077, -77301243, 30931875, -5709042, 0, 1, -72, 2492, -55183, 877812

Offset: 1

Alois P. Heinz, May 04 2012

The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.

			3 example graphs:                          o---o---o
.                                          |\ /|\ /|
.                                          | X | X |
.                                          |/ \|/ \|
.                             o---o        o---o---o
.                             |\ /|        |\ /|\ /|
.                             | X |        | X | X |
.                             |/ \|        |/ \|/ \|
.                o            o---o        o---o---o
Graph:        DG_(1,1)       DG_(2,2)       DG_(3,3)
Vertices:        1              4              9
Edges:           0              6             20
The square diagonal grid graph DG_(2,2) equals the complete graph K_4 and has chromatic polynomial q*(q-1)*(q-2)*(q-3) = q^4 -6*q^3 +11*q^2 -6*q => row 2 = [1, -6, 11, -6, 0].
Triangle T(n,k) begins:
1,    0;
1,   -6,    11,      -6,        0;
1,  -20,   174,    -859,     2627,      -5082, ...
1,  -42,   825,  -10054,    85011,    -528254, ...
1,  -72,  2492,  -55183,   877812,  -10676360, ...
1, -110,  5895, -205054,  5203946, -102687204, ...
1, -156, 11946, -598491, 22059705, -637802510, ...

Alois P. Heinz, Rows n = 1..7, flattened
Wikipedia, Chromatic Polynomial

Columns 1-2 give: A000012, (-1)*A002943(n-1).
Row sums (for n>1) and last elements of rows give: A000004, row lengths give: A002522.
Cf. A000290, A212209.

A229079 Number A(n,k) of ascending runs in {1,...,k}^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 7, 3, 0, 0, 4, 15, 20, 4, 0, 0, 5, 26, 63, 52, 5, 0, 0, 6, 40, 144, 243, 128, 6, 0, 0, 7, 57, 275, 736, 891, 304, 7, 0, 0, 8, 77, 468, 1750, 3584, 3159, 704, 8, 0, 0, 9, 100, 735, 3564, 10625, 16896, 10935, 1600, 9, 0

Offset: 0

Alois P. Heinz, Sep 14 2013

			A(4,1) = 4: [1,1,1,1].
A(3,2) = 20 = 3+3+2+3+2+2+2+3: [1,1,1], [2,1,1], [1,2,1], [2,2,1], [1,1,2], [2,1,2], [1,2,2], [2,2,2].
A(2,3) = 15 = 2+2+2+1+2+2+1+1+2: [1,1], [2,1], [3,1], [1,2], [2,2], [3,2], [1,3], [2,3], [3,3].
A(1,4) = 4 = 1+1+1+1: [1], [2], [3], [4].
Square array A(n,k) begins:
  0, 0,   0,     0,     0,      0,       0,       0, ...
  0, 1,   2,     3,     4,      5,       6,       7, ...
  0, 2,   7,    15,    26,     40,      57,      77, ...
  0, 3,  20,    63,   144,    275,     468,     735, ...
  0, 4,  52,   243,   736,   1750,    3564,    6517, ...
  0, 5, 128,   891,  3584,  10625,   25920,   55223, ...
  0, 6, 304,  3159, 16896,  62500,  182736,  453789, ...
  0, 7, 704, 10935, 77824, 359375, 1259712, 3647119, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000004, A001477, A066373(n+1) for n>0, A229277, A229278, A229279, A229280, A229281, A229282, A229283, A229284.
Rows n=0-10 give: A000004, A001477, A005449, A099721, A229146, A229147, A229148, A229149, A229150, A229151, A229152.
Main diagonal gives A229078.

A:= (n, k)-> `if`(n=0, 0, k^(n-1)*((n+1)*k+n-1)/2):
seq(seq(A(n,d-n), n=0..d), d=0..12);

a[, 0] = a[0, ] = 0; a[n_, k_] := k^(n-1)*((n+1)*k+n-1)/2; Table[a[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 09 2013 *)

A(n,k) = k^(n-1)*((n+1)*k+n-1)/2 for n>0, A(0,k) = 0.

A242249 Number A(n,k) of rooted trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 7, 4, 0, 0, 1, 4, 15, 26, 9, 0, 0, 1, 5, 26, 82, 107, 20, 0, 0, 1, 6, 40, 188, 495, 458, 48, 0, 0, 1, 7, 57, 360, 1499, 3144, 2058, 115, 0, 0, 1, 8, 77, 614, 3570, 12628, 20875, 9498, 286, 0, 0, 1, 9, 100, 966, 7284, 37476, 111064, 142773, 44947, 719, 0

Offset: 0

Alois P. Heinz, May 09 2014

From Vaclav Kotesovec, Aug 26 2014: (Start)
Column k > 0 is asymptotic to c(k) * d(k)^n / n^(3/2), where constants c(k) and d(k) are dependent only on k. Conjecture: d(k) ~ k * exp(1). Numerically:
d(1)   =   2.9557652856519949747148175... (A051491)
d(2)   =   5.6465426162329497128927135... (A245870)
d(3)   =   8.3560268792959953682760695...
d(4)   =  11.0699628777593263124193026...
d(5)   =  13.7856511100846851989303249...
d(6)   =  16.5022088446930015657112211...
d(7)   =  19.2192613290638657575973462...
d(8)   =  21.9366222112987115910888213...
d(9)   =  24.6541883249893084812976812...
d(10)  =  27.3718979186642404090999595...
d(100) = 272.0126359583480733207362718...
d(101) = 274.7309127032967881125015217...
d(200) = 543.8405620978790523837823296...
d(201) = 546.5588426492458787468860222...
d(101)-d(100) = 2.718276744...
d(201)-d(200) = 2.718280551...
(End)

			Square array A(n,k) begins:
  0,  0,    0,     0,      0,      0,       0,       0, ...
  1,  1,    1,     1,      1,      1,       1,       1, ...
  0,  1,    2,     3,      4,      5,       6,       7, ...
  0,  2,    7,    15,     26,     40,      57,      77, ...
  0,  4,   26,    82,    188,    360,     614,     966, ...
  0,  9,  107,   495,   1499,   3570,    7284,   13342, ...
  0, 20,  458,  3144,  12628,  37476,   91566,  195384, ...
  0, 48, 2058, 20875, 111064, 410490, 1200705, 2984142, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 (2018), doi SIGMA 17 (2021)

Columns k=0-10 give: A063524, A000081, A000151, A006964, A052763, A052788, A246235, A246236, A246237, A246238, A246239.
Rows n=0-3 give: A000004, A000012, A001477, A005449.
Lower diagonal gives A242375.
Cf. A255517, A256064.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
      A(d, k), d=divisors(j))*A(n-j, k)*k, j=1..n-1))/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

nn = 10; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; Transpose[ Table[Flatten[ sol = SolveAlways[ 0 == Series[ t[x] - x Product[1/(1 - x^i)^(n a[i]), {i, 1, nn}], {x, 0, nn}], x]; Flatten[{0, Table[a[n], {n, 1, nn}]}] /. sol], {n, 0, nn}]] // Grid (* Geoffrey Critzer, Nov 11 2014 *)
A[n_, k_] := A[n, k] = If[n<2, n, Sum[Sum[d*A[d, k], {d, Divisors[j]}] *A[n-j, k]*k, {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 04 2014, translated from Maple *)

\\ ColGf gives column generating function
ColGf(N,k) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = k/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); x*Ser(A)}
Mat(vector(8, k, concat(0, Col(ColGf(7, k-1))))) \\ Andrew Howroyd, May 12 2018

G.f. for column k: x*Product_{n>=1} 1/(1 - x^n)^(k*A(n,k)). - Geoffrey Critzer, Nov 13 2014

A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 5, 2, 0, 0, 1, 4, 12, 18, 3, 0, 0, 1, 5, 22, 64, 66, 6, 0, 0, 1, 6, 35, 156, 363, 266, 12, 0, 0, 1, 7, 51, 310, 1193, 2214, 1111, 25, 0, 0, 1, 8, 70, 542, 2980, 9748, 14043, 4792, 52, 0, 0, 1, 9, 92, 868, 6273, 30526, 82916, 91857, 21124, 113, 0

Offset: 0

Alois P. Heinz, Feb 24 2015

From Vaclav Kotesovec, Feb 24 2015: (Start)
k    Limit n->infinity A(n,k)^(1/n)
  2.517540352632003890795354598463447277335981266803... = A246169
  5.249032491228170579164952216184309265343086337648... = A246312
  7.969494030514425004826375511986491746399264355846...
 10.688492754969652458452048798468242930479212456958...
 13.407087472537747579787047072702638639945914705837...
 16.125529360448558670505097146631763969697822205298...
 18.843901825822305757579605844910623225182677164912...
 21.562238702430237066018783115405680041128676137631...
 24.280555694806692616578932533497629224907619468796...
26.998860838916733933849490675388336975888308433826...
271.64425688361559470587959030374804709717287744789...
Conjecture: For big k the limit asymptotically approaches k*exp(1).
(End)

			A(3,2) = 5:
  o    o    o    o      o
  |    |    |    |     / \
  1    1    2    2    1   2
  |    |    |    |
  1    2    1    2
Square array A(n,k) begins:
  0,  0,   0,    0,    0,     0,     0, ...
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  1,   5,   12,   22,    35,    51, ...
  0,  2,  18,   64,  156,   310,   542, ...
  0,  3,  66,  363, 1193,  2980,  6273, ...
  0,  6, 266, 2214, 9748, 30526, 77262, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=1-10 give: A004111, A005753, A052757, A052772, A052797, A255518, A255519, A255520, A255521, A255522.
Rows n=0-4 give: A000004, A000012, A001477, A000326, 2*A051662(k-1) for k>0.
Lower diagonal gives A255523.
Cf. A242249, A256068.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
      k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = If[n<2, n, Sum[A[n-j, k]*Sum[k*A[d, k]*d*(-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 4, 1, 6, 0, 0, 0, 0, 4, 0, 5, 7, 0, 0, 0, 0, 4, 0, 1, 1, 8, 0, 0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 0, 0, 4, 0, 0, 0, 16, 6, 10, 0, 0, 0, 0, 4, 0, 0, 0, 32, 5, 7, 11, 0, 0, 0, 0, 4, 0, 0, 0, 80, 1, 1, 1, 12

Offset: 0

Alois P. Heinz, Jun 06 2015

			Square array A(n,k) begins:
  0,  0,  0,  0,  0,   0,   0,   0,    0,    0, ...
  1,  0,  0,  0,  0,   0,   0,   0,    0,    0, ...
  2,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  3,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  4,  4,  4,  4,  4,   4,   4,   4,    4,    4, ...
  5,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  6,  5,  1,  0,  0,   0,   0,   0,    0,    0, ...
  7,  1,  0,  0,  0,   0,   0,   0,    0,    0, ...
  8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, ...
  9,  6,  5,  1,  0,   0,   0,   0,    0,    0, ...

Alois P. Heinz, Antidiagonals n = 0..115, flattened
J. Kovič, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences 15 (2012), Article 12.3.8
Wikipedia, Arithmetic derivative

Columns k=0-10 give: A001477, A003415, A068346, A099306, A258644, A258645, A258646, A258647, A258648, A258649, A258650.
Rows n=0,1,4,8 give: A000004, A000007, A010709, A129150.
Row 15: A090636, Row 28: A090637, Row 63: A090635, Row 81: A129151, Row 128: A369638, Row 1024: A214800, Row 15625: A129152.
Main diagonal gives A185232.
Antidiagonal sums give A258652.
Cf. also A328383.

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
seq(seq(A(n, h-n), n=0..h), h=0..14);

d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[0] = d[1] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[A[n, h-n], {h, 0, 14}, {n, 0, h}] // Flatten (* Jean-François Alcover, Apr 27 2017, translated from Maple *)

A(n,k) = A003415^k(n).

A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

Original entry on oeis.org

1, 5, 8, 20, 24, 40, 48, 80, 88, 120, 120, 160, 168, 240, 240, 320, 288, 312, 360, 480, 384, 408, 528, 640, 616, 520, 648, 960, 840, 816, 960, 1280, 1072, 1440, 1248, 1248, 1368, 1224, 1360, 1920, 1680, 1920, 1848, 1632, 1872, 2640, 2208, 2560, 2384, 3016

Offset: 1

Rémy Sigrist, Nov 29 2017

This sequence is a variant of A256739; both sequences have nice graphical features.
Replacing "SumXOR" by "Sum" in the name leads to the Jordan function J_2 (A007434).
For any sequence f of nonnegative integers with positive indices:
- let x_f be the unique sequence satisfying SumXOR_{d divides n} x_f(d) = f(n) for any n > 0,
- in particular, x_A000027 = A256739 and x_A000290 = a (this sequence),
- also, x_A178910 = A000027 and x_A055895 = A000079,
- see the links section for a gallery of x_f plots for some classic f functions,
- x_f(1) = f(1),
- x_f(p) = f(1) XOR f(p) for any prime p,
- x_A063524 = A008966,
- x_f(n) = SumXOR_{d divides n and n/d is squarefree} f(d) for any n > 0,
- the function x: f -> x_f is a bijection,
- A000004 is the only fixed point of x (i.e. x_f = f if and only if f = A000004),
- for any sequence f, x_{2*f} = 2 * x_f,
- for any sequences g and f, x_{g XOR f} = x_g XOR x_f.
From Antti Karttunen, Dec 29 2017: (Start)
The transform x_f described above could be called "Xor-Moebius transform of f" because of its analogous construction to Möbius transform with A008683 replaced by A008966 and the summation replaced by cumulative XOR.
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..16383
Rémy Sigrist, Colored scatterplot of the first 2^17-1 terms (where the color is function of A087207(n) % 8)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000010 (Euler totient function)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000203 (sigma)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A001157 (sigma_2)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000040 (prime numbers)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A000720 (PrimePi)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A006370 (Collatz map)
Rémy Sigrist, Scatterplot of the first 2^16 terms of x_A005132 (Recamán's sequence)

Cf. A000004, A000010, A000027, A000040, A000079, A000290, A000720, A001157, A003987, A005132, A006370, A007434, A008966, A055895, A063524, A087207, A178910.

Same transform applied to other sequences: A256739 (A000027), A296206 (A256739), A296207 (A227320), A296203 (A000203), A296208 (A005187), A297110 (A006068), A297108 (A248663), A297106 (A156552).

a(n{, f=k->k^2}) = my (v=0); fordiv(n,d,if (issquarefree(n/d), v=bitxor(v,f(d)))); return (v)

a(n) = SumXOR_{d divides n and n/d is squarefree} d^2.

A258850 A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 2, 4, 0, 0, 0, 1, 4, 5, 0, 0, 0, 0, 4, 3, 6, 0, 0, 0, 0, 4, 2, 7, 7, 0, 0, 0, 0, 4, 1, 4, 4, 8, 0, 0, 0, 0, 4, 0, 4, 4, 12, 9, 0, 0, 0, 0, 4, 0, 4, 4, 20, 12, 10, 0, 0, 0, 0, 4, 0, 4, 4, 32, 20, 11, 11, 0, 0, 0, 0, 4, 0, 4, 4, 80, 32, 5, 5, 12

Offset: 0

Alois P. Heinz, Jun 12 2015

			Square array A(n,k) begins:
  0,  0,  0,  0,  0,   0,   0,    0,     0,     0, ...
  1,  0,  0,  0,  0,   0,   0,    0,     0,     0, ...
  2,  1,  0,  0,  0,   0,   0,    0,     0,     0, ...
  3,  2,  1,  0,  0,   0,   0,    0,     0,     0, ...
  4,  4,  4,  4,  4,   4,   4,    4,     4,     4, ...
  5,  3,  2,  1,  0,   0,   0,    0,     0,     0, ...
  6,  7,  4,  4,  4,   4,   4,    4,     4,     4, ...
  7,  4,  4,  4,  4,   4,   4,    4,     4,     4, ...
  8, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...
  9, 12, 20, 32, 80, 208, 512, 2304, 12288, 81920, ...

Alois P. Heinz, Antidiagonals n = 0..31, flattened

Columns k=0-10 give: A001477, A258851, A258852, A258853, A258854, A258855, A258856, A258857, A258858, A258859, A258860.
Rows n=0,1,4,8 give: A000004, A000007, A010709, A258848.
Antidiagonal sums give A258847.
Main diagonal gives A258849.
Cf. A000720, A258975, A259016.

with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
seq(seq(A(n, h-n), n=0..h), h=0..14);

d[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; d[0] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
Table[Table[A[n, h-n], {n, 0, h}], {h, 0, 14}] // Flatten (* Jean-François Alcover, Apr 24 2016, adapted from Maple *)

A(n,k) = A258851^k(n).
A(A259016(n,k),k) = n.
A(A258975(n),n) = 1.

A045896 Denominator of n/((n+1)*(n+2)) = A026741/A045896.

Original entry on oeis.org

1, 6, 6, 20, 15, 42, 28, 72, 45, 110, 66, 156, 91, 210, 120, 272, 153, 342, 190, 420, 231, 506, 276, 600, 325, 702, 378, 812, 435, 930, 496, 1056, 561, 1190, 630, 1332, 703, 1482, 780, 1640, 861, 1806, 946, 1980, 1035, 2162, 1128, 2352, 1225, 2550, 1326, 2756, 1431

Offset: 0

Ralf W. Grosse-Kunstleve, Dec 11 1999

Also period length divided by 2 of pairs (a,b), where a has period 2*n-2 and b has period n.
From Paul Curtz, Apr 17 2014: (Start)
Difference table of A026741/A045896:
   0,      1/6,    1/6,    3/20,     2/15,    5/42, ...
   1/6,      0,  -1/60,   -1/60,    -1/70,   -1/84, ... = 1/6, -A051712/A051713
  -1/6,  -1/60,      0,   1/420,    1/420,   1/504, ...
   3/20,  1/60,  1/420,       0,  -1/2520, -1/2520, ...
  -2/15, -1/70, -1/420, -1/2520,        0, 1/13860, ...
   5/42,  1/84,  1/504,  1/2520, -1/13860,       0, ...
Autosequence of the first kind. The main diagonal is A000004. The first two upper diagonals are equal. Their denominators are A000911. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Ralf W. Grosse-Kunstleve, Origin of EIS sequences A045895 & A045896. [Wayback Machine copy]
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.

Factor of A160466.

Cf. A000004, A000384, A000911, A045895, A026741, A051714, A241269.

import Data.Ratio ((%), denominator)
a045896 n = denominator $ n % ((n + 1) * (n + 2))
-- Reinhard Zumkeller, Dec 12 2011

seq((n+1)*(n+2)*(3-(-1)^n)/4, n=0..20); # C. Ronaldo
with(combinat): seq(lcm(n+1,binomial(n+2,n)), n=0..50); # Zerinvary Lajos, Apr 20 2008

Table[LCM[2*n + 2, n + 2]/2, {n, 0, 40}] (* corrected by Amiram Eldar, Sep 14 2022 *)
Denominator[#[[1]]/(#[[2]]#[[3]])&/@Partition[Range[0,60],3,1]] (* Harvey P. Dale, Aug 15 2013 *)

Vec((2*x^3+3*x^2+6*x+1)/(1-x^2)^3+O(x^99)) \\ Charles R Greathouse IV, Mar 23 2016

G.f.: (2*x^3+3*x^2+6*x+1)/(1-x^2)^3.
a(n) = (n+1)*(n+2) if n odd; or (n+1)*(n+2)/2 if n even = (n+1)*(n+2)*(3-(-1)^n)/4. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
a(2*n) = A000384(n+1); a(2*n+1) = A026741(n+1). - Reinhard Zumkeller, Dec 12 2011
Sum_{n>=0} 1/a(n) = 1 + log(2). - Amiram Eldar, Sep 11 2022
From Amiram Eldar, Sep 14 2022: (Start)
a(n) = lcm(2*n+2, n+2)/2.
a(n) = A045895(n+2)/2. (End)
E.g.f.: (2 + 8*x + x^2)*cosh(x)/2 + (2 + 2*x + x^2)*sinh(x). - Stefano Spezia, Apr 24 2024

A210485 Number T(n,k) of parts in all partitions of n in which no part occurs more than k times; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 3, 3, 6, 0, 3, 8, 8, 12, 0, 5, 11, 15, 15, 20, 0, 8, 17, 24, 29, 29, 35, 0, 10, 23, 36, 41, 47, 47, 54, 0, 13, 36, 50, 65, 71, 78, 78, 86, 0, 18, 48, 75, 91, 104, 111, 119, 119, 128, 0, 25, 69, 102, 132, 150, 165, 173, 182, 182, 192

Offset: 0

Alois P. Heinz, Jan 23 2013

T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A006128(n) for k >= n.

For fixed k > 0, T(n,k) ~ 3^(1/4) * log(k+1) * exp(Pi*sqrt(2*k*n/(3*(k+1)))) / (Pi * (8*k*(k+1)*n)^(1/4)). - Vaclav Kotesovec, Oct 18 2018

			T(6,2) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
Triangle T(n,k) begins:
  0;
  0,  1;
  0,  1,  3;
  0,  3,  3,  6;
  0,  3,  8,  8, 12;
  0,  5, 11, 15, 15, 20;
  0,  8, 17, 24, 29, 29, 35;
  0, 10, 23, 36, 41, 47, 47, 54;
  0, 13, 36, 50, 65, 71, 78, 78, 86;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000004, A015723, A185350, A117148, A320607, A320608, A320609, A320610, A320611, A320612, A320613.
Main diagonal gives A006128.
T(2n,n) gives A364245.
Cf. A213177, A286653.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = Sum_{i=0..k} A213177(n,i).

cp's OEIS Frontend

A195152 Square array read by antidiagonals with T(n,k) = n*((k+2)*n-k)/2, n=0, +- 1, +- 2,..., k>=0.

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A212208 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square diagonal grid graph DG_(n,n), highest powers first.

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A229079 Number A(n,k) of ascending runs in {1,...,k}^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A242249 Number A(n,k) of rooted trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A258651 A(n,k) = n^(k) = k-th arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A295901 Unique sequence satisfying SumXOR_{d divides n} a(d) = n^2 for any n > 0, where SumXOR is the analog of summation under the binary XOR operation.

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A258850 A(n,k) = k-th pi-based arithmetic derivative of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A195152 Square array read by antidiagonals with T(n,k) = n((k+2)n-k)/2, n=0, +- 1, +- 2,..., k>=0.