A338586 -id:A338586 - 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).

A338778 Number of ordered ways of writing n-th tetrahedral number as a sum of n positive tetrahedral numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 20, 195, 1890, 6286, 94584, 1065120, 12345432, 194450586, 2844976135, 44569913570, 740023110855, 13144353701940, 241663182769494, 4707408836458200, 95865898167054186, 2038122531703155798, 45103282424247100962, 1037559653596650520776, 24776005985596646165127

Offset: 0

Ilya Gutkovskiy, Nov 08 2020

nonn

			The 5th tetrahedral number is 35 and 35 = 1 + 4 + 10 + 10 + 10 (20 permutations), so a(5) = 20.

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for sequences related to compositions

Cf. A000292, A298269, A298672, A298857, A298858, A303170, A338586.

a(n) = [x^A000292(n)] (Sum_{j>=1} x^A000292(j))^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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A338585 Number of partitions of the n-th triangular number into exactly n positive triangular numbers.

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A338778 Number of ordered ways of writing n-th tetrahedral number as a sum of n positive tetrahedral numbers.

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

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