A166444 -id:A166444 - cp's OEIS frontend

A165426 a(1) = 1, a(2) = 8, a(n) = product of the previous terms for n >= 3.

1, 8, 8, 64, 4096, 16777216, 281474976710656, 79228162514264337593543950336, 6277101735386680763835789423207666416102355444464034512896

Offset: 1

Jaroslav Krizek, Sep 17 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..13

a[1]:= 1; a[2]:= 8; a[n_]:= Product[a[j], {j,1,n-1}]; Table[a[n],{n,1, 12}] (* G. C. Greubel, Oct 19 2018 *)

{a(n) = if(n==1, 1, if(n==2, 8, prod(j=1,n-1, a(j))))};
for(n=1,10, print1(a(n), ", ")) \\ G. C. Greubel, Oct 19 2018

a(1) = 1, a(2) = 8, a(n) = Product_{i=1..n-1} a(i), n >= 3.
a(1) = 1, a(2) = 8, a(n) = A001018(2^(n-3)) = 8^(2^(n-3)), n >= 3.
a(1) = 1, a(2) = 8, a(3) = 8, a(n) = (a(n-1))^2, n >= 4.
a(n) = 8^A166444(n). [uncovered by sequencedb.net]. - R. J. Mathar, Jun 30 2021

A339419 Number of compositions (ordered partitions) of n into an odd number of squares.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 3, 1, 5, 5, 7, 14, 10, 27, 27, 44, 69, 73, 144, 158, 260, 366, 466, 775, 940, 1490, 2031, 2803, 4264, 5551, 8460, 11525, 16399, 23864, 32435, 47981, 66005, 94701, 135072, 187999, 272678, 379095, 543626, 769490, 1083788, 1553661, 2177681, 3113333

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(9) = 5 because we have [9], [4, 4, 1], [4, 1, 4], [1, 4, 4] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A006456, A166444, A317665, A339365, A339417, A339418.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

nmax = 47; CoefficientList[Series[1/(3 - EllipticTheta[3, 0, x]) - 1/(1 + EllipticTheta[3, 0, x]), {x, 0, nmax}], x]

G.f.: 1 / (3 - theta_3(x)) - 1 / (1 + theta_3(x)), where theta_3() is the Jacobi theta function.
a(n) = (A006456(n) - A317665(n)) / 2.
a(n) = -Sum_{k=0..n-1} A006456(k) * A317665(n-k).

A192933 Triangle read by rows: T(n,k) = Sum_{i <= n, j <= k, (i,j) <> (n,k)} T(i,j), starting with T(1,1) = 1, for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 2, 6, 12, 4, 16, 44, 88, 8, 40, 136, 360, 720, 16, 96, 384, 1216, 3152, 6304, 32, 224, 1024, 3712, 11296, 28896, 57792, 64, 512, 2624, 10624, 36416, 108032, 273856, 547712, 128, 1152, 6528, 29056, 109696, 362624, 1056896, 2661504, 5323008, 256, 2560, 15872, 76800, 314880, 1135616, 3659776, 10528768, 26380544, 52761088

Offset: 1

Andrea Raffetti, Jul 13 2011

The outer diagonal is A059435.
The second outer diagonal is A090442.
The third outer diagonal is essentially 2*A068766.
The first column is A011782.
The second column is essentially A057711 (not considering its first two terms).
The second column is essentially A129952 (not considering its first two terms).
The second column is essentially 2*A001792.
The differences between the terms of the second column is essentially 2*A045623.
The third column is essentially 4*A084266.
The cumulative sums of the third column are essentially 4*A176027.
T(n,k) = 0 for n < k. If this overriding constraint is not applied, you get A059576. - Franklin T. Adams-Watters, Jul 24 2011
For n >= 2 and 1 <= k <= n, T(n,k) is the number of bimonotone subdivisions of a 2-row grid with n points on the first row and k points on the second row (with the lower left point of the grid being the origin). A bimonotone subdivision of a convex polygon (the convex hull of the grid) is one where the internal dividing lines have nonnegative (including infinite) slopes. See Robeva and Sun (2020). - Petros Hadjicostas and Michel Marcus, Jul 15 2020

			Triangle (with rows n >= 1 and columns k = 1..n) begins:
   1;
   1,   2;
   2,   6,   12;
   4,  16,   44,    88;
   8,  40,  136,   360,   720;
  16,  96,  384,  1216,  3152,   6304;
  32, 224, 1024,  3712, 11296,  28896,  57792;
  64, 512, 2624, 10624, 36416, 108032, 273856, 547712;
  ...
Example: T(4,3) = 44 = 1 + 1 + 2 + 2 + 6 + 12 + 4 + 16.
From _Petros Hadjicostas_, Jul 15 2020: (Start)
Consider the following 2-row grid with n = 3 points at the top and k = 2 points at the bottom:
   A  B  C
   *--*--*
   |    /
   |   /
   *--*
   D  E
The sets of the dividing internal lines of the T(3,2) = 6 bimonotone subdivisions of the above 2-row grid are as follows: { }, {DC}, {DB}, {EB}, {DB, DC}, and {DB, EB}. We exclude subdivisions {EA} and {EA, EB} because they have at least one dividing line with a negative slope. (End)

Andrea Raffetti, Rows n = 1..13 of triangle
Elina Robeva and Melinda Sun, Bimonotone Subdivisions of Point Configurations in the Plane, arXiv:2007.00877 [math.CO], 2020.

Cf. A001003, A001792, A006318, A011782, A045623, A057711, A059435, A059576, A068766, A084266, A090442, A129952, A166444, A176027, A336245.

lista(nn) = {my(T=matrix(nn, nn)); T[1,1] = 1; for (n=2, nn, for (k=1, n, T[n,k] = sum(i=1, n, sum(j=1, k, if ((i!=n) || (j!=k), T[i,j]))););); vector(nn, k, vector(k, i, T[k, i]));} \\ Michel Marcus, Mar 18 2020

T(n,1) = 2^(n-2) for n >= 2.
T(n,2) = n*2^(n-2) for n >= 2.
T(n,3) = 2^(n-2)*((n-k+1)^2 + 7*(n-k+1) + 4)/2 = 2^(n-3)*(n^2 + 3*n - 6) for k = 3 and n >= 3.
In general: For 1 <= k <= n with (n,k) <> 1,
T(n,k) = 2^(n-2)*Sum_{i=0..k-1} c(k,i)*(n-k+1)^(k-1-i)/(k-1)! and
T(n,k) = 2^(n-2)*Sum_{j=0..k-1} c(k,k-1-j)*(n-k+1)^j/(k-1)!
with c(k,i) being specific coefficients. Below are the first values for c(k,i):
  1;
  1,  1;
  1,  7,    4;
  1, 18,   77,   36;
  1, 34,  359, 1238,   528,
  1, 55, 1065, 8705, 26654, 10800;
... [Formula corrected by Petros Hadjicostas, Jul 15 2020]
The diagonal of this triangle for c(k,i) divided by (k-1)! (except for the first term) is equal to the Shroeder number sequence A006318(k-1).
From Petros Hadjicostas and Michel Marcus, Jul 15 2020: (Start)
T(n,1) = 2^(n-2) for n >= 2; T(n,k) = 2*(T(n,k-1) + T(n-1,k) - T(n-1,k-1)) for n > k >= 2; T(n,n) = 2*T(n,n-1) for n = k >= 2; and T(n,k) = 0 for 1 <= n < k. [Robeva and Sun (2020)] (They do not specify T(1,1) explicitly since they do not care about subdivisions of a degenerate polygon with only one side.)
T(n,k) = (2^(n-2)/(k-1)!) * P_k(n) = (2^(n-2)/(k-1)!) * Sum_{j=1..k} A336245(k,j)*n^(k-j) for n >= k >= 1 with (n,k) <> (1,1), where P_k(n) is some polynomial with integer coefficients of degree k-1. [Robeva and Sun (2020)]
A336245(k,j) = Sum_{s=0..j-1} c(k,s) * binomial(k-1-s, k-j) * (1-k)^(j-1-s) for 1 <= j <= k, in terms of the above coefficients c(k,i).
So c(k,s) = Sum_{j=1..s+1} A336245(k,j) * binomial(k-j, k-s-1) * (k-1)^(s+1-j) for k >= 1 and 0 <= s <= k-1, obtained by inverting the binomial transform.
Bivariate o.g.f.: x*y*(1 - x)*(1 - 2*y*g(2*x*y))/(1 - 2*x - 2*y + 2*x*y), where g(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = g.f. of A001003.
Letting y = 1 in the above joint o.g.f., we get the o.g.f. of the row sums: x*(1-x)*(2*g(2*x) - 1). It can then be easily proved that
Sum_{k=1..n} T(n,k) = 2^n*A001003(n-1) - 2^(n-1)*A001003(n-2) for n >= 3. (End)

Offset changed by Andrew Howroyd, Dec 31 2017

Name edited by Petros Hadjicostas, Jul 15 2020

A339417 Number of compositions (ordered partitions) of n into an odd number of triangular numbers.

Original entry on oeis.org

0, 1, 0, 2, 0, 4, 1, 9, 3, 19, 12, 41, 33, 91, 92, 203, 238, 466, 602, 1080, 1493, 2536, 3661, 6001, 8902, 14278, 21554, 34094, 52013, 81602, 125297, 195582, 301475, 469193, 724881, 1126161, 1742206, 2703888, 4186276, 6493192, 10057553, 15594636, 24161364, 37455851

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(8) = 3 because we have [6, 1, 1], [1, 6, 1] and [1, 1, 6].

Cf. A000217, A023361, A106507, A166444, A339374, A339416.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+g, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

nmax = 43; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}]) - 1/Sum[x^(k (k + 1)/2), {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k*(k + 1)/2)) - 1 / Sum_{k>=0} x^(k*(k + 1)/2)).
a(n) = (A023361(n) - A106507(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023361(k) * A106507(n-k).

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).

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

A339409 Number of compositions (ordered partitions) of n into an odd number of primes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 4, 3, 4, 7, 12, 19, 22, 32, 53, 80, 120, 160, 245, 368, 553, 800, 1164, 1736, 2588, 3813, 5598, 8226, 12228, 18060, 26657, 39221, 57945, 85656, 126506, 186584, 275307, 406514, 600488, 886255, 1308088, 1930648, 2850861, 4208743, 6212824, 9170440, 13538025

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(8) = 3 because we have [3, 3, 2], [3, 2, 3] and [2, 3, 3].

Index entries for sequences related to compositions

Cf. A000040, A023360, A166444, A184199, A339381, A339408.

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-ithprime(j), 1-t), j=1..numtheory[pi](n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 03 2020

nmax = 47; CoefficientList[Series[(1/2) (1/(1 - Sum[x^Prime[k], {k, 1, nmax}]) - 1/(1 + Sum[x^Prime[k], {k, 1, nmax}])), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^prime(k)) - 1 / (1 + Sum_{k>=1} x^prime(k))).

A339421 Number of compositions (ordered partitions) of n into an odd number of cubes.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 4, 11, 11, 13, 22, 15, 37, 18, 56, 29, 80, 56, 109, 107, 142, 190, 184, 313, 255, 490, 391, 731, 644, 1045, 1082, 1458, 1792, 2044, 2895, 2957, 4531, 4463, 6863, 6972, 10126, 11090, 14739, 17691, 21484, 27954, 31741

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(10) = 3 because we have [8, 1, 1], [1, 8, 1] and [1, 1, 8].

Cf. A000578, A023358, A166444, A323633, A339369, A339419, A339420.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+3*g*(g-1)+1, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020

nmax = 57; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]) - 1/Sum[x^(k^3), {k, 0, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k^3)) - 1 / Sum_{k>=0} x^(k^3)).
a(n) = (A023358(n) - A323633(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023358(k) * A323633(n-k).

A346634 Number of strict odd-length integer partitions of 2n + 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937

Offset: 0

Gus Wiseman, Aug 01 2021

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
                 (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
                          (5,3,1)  (6,3,2)  (6,5,2)   (7,5,3)
                          (6,2,1)  (6,4,1)  (7,4,2)   (7,6,2)
                                   (7,3,1)  (7,5,1)   (8,4,3)
                                   (8,2,1)  (8,3,2)   (8,5,2)
                                            (8,4,1)   (8,6,1)
                                            (9,3,1)   (9,4,2)
                                            (10,2,1)  (9,5,1)
                                                      (10,3,2)
                                                      (10,4,1)
                                                      (11,3,1)
                                                      (12,2,1)
                                                      (5,4,3,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Odd bisection of A067659, which is ranked by A030059.
The even version is the even bisection of A067661.
The case of all odd parts is counted by A069911 (non-strict: A078408).
The non-strict version is A160786, ranked by A340931.
The non-strict even version is A236913, ranked by A340784.
The even-length version is A343942 (non-strict: A236914).
The even-sum version is A344650 (non-strict: A236559 or A344611).
A000009 counts partitions with all odd parts, ranked by A066208.
A000009 counts strict partitions, ranked by A005117.
A027193 counts odd-length partitions, ranked by A026424.
A027193 counts odd-maximum partitions, ranked by A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Other cases of odd length:
- A024429 set partitions
- A089677 ordered set partitions
- A166444 compositions
- A174726 ordered factorizations
- A332304 strict compositions
- A339890 factorizations
Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]

More terms from Alois P. Heinz, Aug 05 2021

A348396 Number of ways to reach n by starting with 1 and repeatedly adding any positive integer or multiplying by any integer greater than 1.

Original entry on oeis.org

1, 2, 4, 10, 18, 42, 78, 168, 328, 672, 1324, 2706, 5354, 10788, 21518, 43194, 86208, 172792, 345208, 691118, 1381616, 2764476, 5527626, 11058184, 22113454, 44232246, 88459468, 176929482, 353848086, 707718428, 1415414600, 2830872574, 5661703102, 11323491086

Offset: 1

Michael R Peake, Jan 25 2022

			For n = 3 the a(3) = 4 solutions are 1 + 2, (1 + 1) + 1, (1*2) + 1, 1*3.

Alois P. Heinz, Table of n, a(n) for n = 1..3322

Cf. A074206, A166444, A348378.

a(1)=1; for n=2:20, a(n)=sum(a(1:n-1))+sum(a(find(~rem(n,1:n-1)))); end;

a:= proc(n) option remember; uses numtheory; `if`(n=1, 1,
      add(a(n-j), j=1..n-1)+add(a(n/d), d=divisors(n) minus {1}))
    end:
seq(a(n), n=1..34);  # Alois P. Heinz, Jan 25 2022

a[n_] := a[n] = If[n == 1, 1, Sum[a[n - j], {j, 1, n - 1}] +
     Sum[a[n/d], {d, Divisors[n] ~Complement~ {1}}]];
Table[a[n], {n, 1, 34}] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

seq(n)={my(a=vector(n), s=1); a[1]=s; for(n=2, n, a[n] = s + sumdiv(n, d, a[d]); s += a[n]); a} \\ Andrew Howroyd, Jan 25 2022

from functools import cache
@cache
def a(n): return 1 if n == 1 else 1 + sum(a(i) for i in range(1, n)) + sum(a(i) for i in range(2, n) if n%i == 0)
print([a(n) for n in range(1, 34)]) # Michael S. Branicky, Jan 25 2022

cp's OEIS Frontend

A165426 a(1) = 1, a(2) = 8, a(n) = product of the previous terms for n >= 3.

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A339419 Number of compositions (ordered partitions) of n into an odd number of squares.

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A192933 Triangle read by rows: T(n,k) = Sum_{i <= n, j <= k, (i,j) <> (n,k)} T(i,j), starting with T(1,1) = 1, for n >= 1 and 1 <= k <= n.

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A339417 Number of compositions (ordered partitions) of n into an odd number of triangular numbers.

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

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A339409 Number of compositions (ordered partitions) of n into an odd number of primes.

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A339421 Number of compositions (ordered partitions) of n into an odd number of cubes.

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