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A338585
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Number of partitions of the n-th triangular number into exactly n positive triangular numbers.
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3
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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
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OFFSET
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0,6
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LINKS
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FORMULA
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EXAMPLE
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The 5th triangular number is 15 and 15 = 1 + 1 + 1 + 6 + 6 = 3 + 3 + 3 + 3 + 3, so a(5) = 2.
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MAPLE
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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*k<n or k>n, 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):
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MATHEMATICA
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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];
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PROG
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(SageMath) # Returns a list of length n, slow.
def GeneralizedEulerTransform(n, a):
R.<x, y> = 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)
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CROSSREFS
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Cf. A000217, A007294, A072964, A106337, A196010, A288126, A298858, A307614, A319435, A319797, A319799, A331900, A338586.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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