A072964 -id:A072964 - cp's OEIS frontend

A338585 Number of partitions of the n-th triangular number into exactly n positive triangular numbers.

1, 1, 0, 0, 1, 2, 3, 4, 9, 16, 29, 52, 92, 173, 307, 554, 1002, 1792, 3216, 5738, 10149, 17942, 31769, 55684, 97478, 170356, 295644, 512468, 886358, 1523779, 2614547, 4476152, 7627119, 12966642, 21988285, 37142199, 62591912, 105215149, 176266155, 294591431

Offset: 0

Ilya Gutkovskiy, Nov 08 2020

nonn

			The 5th triangular number is 15 and 15 = 1 + 1 + 1 + 6 + 6 = 3 + 3 + 3 + 3 + 3, so a(5) = 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers
Index entries for sequences related to partitions

Cf. A000217, A007294, A072964, A106337, A196010, A288126, A298858, A307614, A319435, A319797, A319799, A331900, A338586.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(i*kn, 0, b(n, h(i-1), k)+b(n-i, h(min(n-i, i)), k-1)))
    end:
a:= n-> (t-> b(t, h(t), n))(n*(n+1)/2):
seq(a(n), n=0..42);  # Alois P. Heinz, Nov 10 2020

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ@Sqrt[8n+1], n, h[n-1]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0], If[i k < n || k > n, 0, b[n, h[i-1], k] + b[n-i, h[Min[n-i, i]], k-1]]];
a[n_] := b[#, h[#], n]&[n(n+1)/2];
a /@ Range[0, 42](* Jean-François Alcover, Nov 15 2020, after Alois P. Heinz *)

# Returns a list of length n, slow.
def GeneralizedEulerTransform(n, a):
    R. = ZZ[[]]
    f = prod((1 - y*x^a(k) + O(x, y)^a(n)) for k in (1..n))
    coeffs = f.inverse().coefficients()
    coeff = lambda k: coeffs[x^a(k)*y^k] if x^a(k)*y^k in coeffs else 0
    return [coeff(k) for k in range(n)]
def A338585List(n): return GeneralizedEulerTransform(n, lambda n: n*(n+1)/2)
print(A338585List(12)) # Peter Luschny, Nov 12 2020

a(n) = [x^A000217(n) y^n] Product_{j>=1} 1 / (1 - y*x^A000217(j)).

a(n) = A319797(A000217(n),n).

A299031 Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers.

Original entry on oeis.org

1, 1, 0, 3, 18, 60, 252, 1576, 10494, 64152, 458400, 3407019, 27713928, 225193982, 1980444648, 17626414158, 165796077562, 1593587604441, 15985672426992, 163422639872978, 1729188245991060, 18743981599820280, 208963405365941380, 2378065667103672024, 27742569814633730608

Offset: 0

Ilya Gutkovskiy, Feb 01 2018

nonn

			a(3) = 3 because third triangular number is 6 and we have [4, 1, 1], [1, 4, 1] and [1, 1, 4].

Cf. A000217, A000290, A066535, A072964, A104383, A126683, A196010, A224677, A224679, A278340, A288126, A298329, A298858, A298938, A299032.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n (n + 1)/2}], {n, 0, 24}]

a(n) = [x^(n*(n+1)/2)] (Sum_{k>=0} x^(k^2))^n.

A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 3, 13, 3, 55, 201, 159, 865, 1803, 7093, 43431, 14253, 22903, 130851, 120763, 1099693, 4527293, 4976767, 7516897, 14349685, 72866239, 81946383, 167841291, 897853735, 455799253, 946267825, 5054280915, 3941268001, 17066300985, 49111862599

Offset: 0

Ilya Gutkovskiy, Jan 31 2020

nonn

			a(6) = 3 because we have [21], [15, 6] and [6, 15].

Alois P. Heinz, Table of n, a(n) for n = 0..300
Index entries for sequences related to compositions

Cf. A000217, A072964, A126683, A196010, A224677, A288126, A298858, A307614, A331843, A331884.

b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n*(n+1)/2, n, 0):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 31 2020

b[n_, i_, p_] := b[n, i, p] = With[{t = i(i+1)/2}, If[t(i+2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t > n, 0, b[n-t, i-1, p+1]]]]];
a[n_] := b[n(n+1)/2, n, 0];
a /@ Range[0, 37] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

a(n) = A331843(A000217(n)).

More terms from Alois P. Heinz, Jan 31 2020

A114738 Triangle read by rows: row n lists partitions of n-th triangular number into triangular parts.

Original entry on oeis.org

1, 3, 111, 6, 33, 3111, 111111, 10, 631, 61111, 3331, 331111, 31111111, 1111111111, 15, 10311, 1011111, 663, 66111, 6333, 633111, 63111111, 6111111111, 33333, 3333111, 333111111, 33111111111, 3111111111111, 111111111111111, 21, 156

Offset: 1

Amarnath Murthy, Nov 15 2005

			1
3,111
6,33,3111,111111
10,631,61111,3331,331111,31111111,1111111111
15,10311,1011111,663,66111,6333,633111,63111111,6111111111,33333,3333111,333111111,33111111111,3111111111111,111111111111111.
...

Cf. A072964.

A306597 a(n) = Card({ Sum_{k=1..n}(x_k * k) : (x_k){k=1..n} is an n-tuple of nonnegative integers such that Sum{k=1..n}(x_k * T_k) = T_n }), where T_k denotes the k-th triangular number.

Original entry on oeis.org

1, 2, 4, 6, 9, 15, 20, 27, 34, 43, 52, 63, 75, 87, 102, 117, 132, 149, 166, 185, 206, 226, 248, 271, 294, 318, 345, 373, 399, 429, 459, 489, 520, 554, 587, 623, 658, 695, 734, 772, 811, 853, 894, 936, 981, 1026, 1072, 1119, 1167, 1215, 1266, 1316, 1368, 1420

Offset: 1

Luc Rousseau, Feb 27 2019

nonn

Inspired by the questions:
- Q1: into how many regions do n+1 straight lines divide the plane?
- Q2: what is the number of possible answers to Q1?
This sequence provides an answer to an analog of Q2 in a modified version of the problem.
Also an analog of A069999(n) with the roles of k and T_k swapped in the definition.

			When n = 3, n*(n+1)/2 = 6. All possible ways to partition 6 into parts with triangular sizes (1, 3, 6) are:
  0*1 + 0*3 + 1*6 = 6
  0*1 + 2*3 + 0*6 = 6
  3*1 + 1*3 + 0*6 = 6
  6*1 + 0*3 + 0*6 = 6
In the above products, keep the left multiplicands and replace the right ones with their triangular roots:
  0*1 + 0*2 + 1*3 = 3
  0*1 + 2*2 + 0*3 = 4
  3*1 + 1*2 + 0*3 = 5
  6*1 + 0*2 + 0*3 = 6
Card({ 3, 4, 5, 6 }) = 4, so a(3) = 4.

Luc Rousseau, Table of n, a(n) for n = 1..150
Luc Rousseau, C program
Luc Rousseau, Java program (standalone version)
Luc Rousseau, Java program (jOEIS version, github)

Cf. A177862, A072964, A069999.

T[n_] := n*(n + 1)/2
R[n_] := (Sqrt[8*n + 1] - 1)/2
S[0] := 0
S[d_] := S[d] =
  Module[{r = R[d]},
   If[IntegerQ[r], r++; r + T[r],
    First@TakeSmallest[
       1]@(S[#[[1]]] + S[#[[2]]] & /@ IntegerPartitions[d, {2}])]]
A0[n_] := Sum[Boole[d + S[d] <= 2*n], {d, 0, n}]
A[n_] := A0[T[n]]
For[n = 1, n <= 150, n++, Print[n, " ", A[n]]]

cp's OEIS Frontend

A338585 Number of partitions of the n-th triangular number into exactly n positive triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A299031 Number of ordered ways of writing n-th triangular number as a sum of n squares of nonnegative integers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A114738 Triangle read by rows: row n lists partitions of n-th triangular number into triangular parts.

Views

Author

Keywords

Examples

Crossrefs

A306597 a(n) = Card({ Sum_{k=1..n}(x_k * k) : (x_k){k=1..n} is an n-tuple of nonnegative integers such that Sum{k=1..n}(x_k * T_k) = T_n }), where T_k denotes the k-th triangular number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica