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A292072
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Main diagonal of A292068.
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2
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1, -1, -3, -20, 66, 4439, 454420, 4873175, -3803048954, -7320203267692, -1403057989033446, 6669491545211096686, 78492109668913945526447, 69591502229308312804788424, -6243846072108996200105800383026, -604234376454072219680822138902122079
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^n] Product_{k=1..n} 1/(1 + k^n*x^k).
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MAPLE
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b:= proc(n, i, k) option remember; (m->
`if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
`if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
end:
g:= proc(n, k) option remember; `if`(n=0, 1,
-add(b(n-i$2, k)*g(i, k), i=0..n-1))
end:
a:= n-> g(n$2):
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = Function[m, If[m < n, 0, If[n == m, i!^k, b[n, i - 1, k] + If[i > n, 0, i^k*b[n - i, i - 1, k]]]]][i*(i + 1)/2];
g[n_, k_] := g[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k]*g[i, k], {i, 0, n-1}]];
a[n_] := g[n, n];
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PROG
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(PARI) {a(n) = polcoeff(1/prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}
(Python)
from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, k):
m=i*(i + 1)/2
return 0 if m<n else f(i)**k if n==m else b(n, i - 1, k) + (0 if i>n else i**k*b(n - i, i - 1, k))
@cacheit
def g(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*g(i, k) for i in range(n)])
def a(n): return g(n, n)
print([a(n) for n in range(16)]) # Indranil Ghosh, Sep 14 2017, after Maple program
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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