A319799 -id:A319799 - cp's OEIS frontend

A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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

Offset: 0

Alois P. Heinz, Sep 28 2018

Equals A181506 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 1;
  0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A010054 (for n>0), A052344, A063993, A319814, A319815, A319816, A319817, A319818, A319819, A319820.
Row sums give A007294.
T(2n,n) gives A319799.
Cf. A000217, A117278, A181506, A243148, A319394.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ @ Sqrt[8*n + 1], n, h[n - 1]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[ x*b[n - i, h[Min[n - i, i]]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, h[n]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

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

A111178 Number of partitions of n into positive numbers one less than a square.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 2, 5, 2, 4, 5, 2, 5, 5, 2, 6, 7, 4, 6, 7, 5, 6, 8, 6, 8, 12, 6, 9, 13, 6, 10, 15, 8, 14, 15, 9, 16, 16, 10, 18, 21, 14, 19, 22, 16, 20, 24, 19, 25, 30, 20, 27, 33, 21, 29, 39, 26, 37, 40, 28, 42, 42, 31, 48

Offset: 0

Wouter Meeussen, Oct 22 2005

Also limiting form of the number of representations of n into k positive squares for k decreasing from n to 1, or Table[Count[SumOfSquaresRepresentations[k,n], {a_,}/;a>0], {n,100,100}, {k,100,40,-1}]. (Franklin T. Adams-Watters: replacing k^2 ones by the value k^2 changes the count by k^2-1).

a(n) = A243148(2n,n). - Alois P. Heinz, May 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

Cf. A001156, A078134.

Cf. A005563, A319799.

a111178 = p $ tail a005563_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Apr 02 2014

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)+
      `if`(i^2>n+1, 0, b(n+1-i^2, i))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, May 30 2014

nn = 100; CoefficientList[Series[Product[1/(1 - x^(k^2 - 1)), {k, 2, nn}], {x, 0, nn}], x] (* corrected by T. D. Noe, Feb 22 2012 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i^2>n+1, 0, b[n+1-i^2, i]]]]; a[n_] := b[n, Round[Sqrt[n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

G.f.: Product_{k>=2} 1/(1-x^(k^2-1)).

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

Original entry on oeis.org

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).

A338465 Number of ways to write 2*n as an ordered sum of n nonzero triangular numbers.

Original entry on oeis.org

1, 0, 2, 0, 6, 5, 20, 42, 70, 261, 297, 1430, 1584, 7293, 9634, 35945, 60150, 176596, 366401, 886977, 2150421, 4624410, 12205074, 25065216, 67616872, 139894305, 369551925, 793214982, 2011977414, 4517758504, 10992821055, 25669627965, 60531471286, 145112506352

Offset: 0

Ilya Gutkovskiy, Jan 31 2021

nonn

Also number of ways to write n as an ordered sum of n nonnegative numbers one less than a triangular number.

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

Cf. A000096, A000217, A106337, A298858, A319799.

Table[SeriesCoefficient[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^n, {x, 0, 2 n}], {n, 0, 33}]

a(n) = [x^(2*n)] (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^n, where theta_2() is the Jacobi theta function.

a(n) = [x^n] (Sum_{k>=0} x^(k*(k + 3)/2))^n.

A341773 Number of partitions of 2*n into exactly n nonzero tetrahedral numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 1, 0, 3, 1, 0, 3, 1, 0, 4, 2, 0, 4, 2, 0, 4, 3, 0, 5, 4, 1, 5, 4, 1, 5, 5, 1, 6, 6, 2, 6, 6, 2, 6, 7, 3, 7, 9, 4, 8, 9, 4, 8, 10, 5, 9, 12, 6, 10, 12, 7, 10, 13, 8, 12, 15, 10, 13, 16, 11, 13, 17, 12

Offset: 0

Ilya Gutkovskiy, Feb 19 2021

nonn

David A. Corneth, Table of n, a(n) for n = 0..2999

Cf. A000292, A024692, A062748, A068980, A298269, A319799, A338586, A341774, A341775, A341776, A341777, A341778.

nmax = 80; CoefficientList[Series[Product[1/(1 - x^(Binomial[k + 4, 3] - 1)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1 / (1 - x^(binomial(k+4,3)-1)).

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A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A111178 Number of partitions of n into positive numbers one less than a square.

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

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A338465 Number of ways to write 2*n as an ordered sum of n nonzero triangular numbers.

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A341773 Number of partitions of 2*n into exactly n nonzero tetrahedral numbers.

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