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1, 5, 51, 747, 13245, 264329, 5721415, 131425079, 3159389817, 78729848397, 2019910325499, 53087981674275, 1423867359013749, 38855956977763857, 1076297858301372687, 30203970496501504239, 857377825323716359665, 24586286492003180067989, 711463902659879056604995, 20756358426519694831851227
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OFFSET
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1,2
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COMMENTS
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Conjecture: (i) All the terms are odd integers. Also, p | a(p) for any odd prime p.
(ii) Let D_n(x) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*x^k = Sum_{k=0..n} binomial(n,k)^2*x^k*(x+1)^(n-k) for n >= 0, and s_n(x) = Sum_{k=1..n} (binomial(n,k)*binomial(n,k-1)/n)*x^(k-1)*(x+1)^(n-k) = (Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*x^k/(k+1))/(x+1) for n > 0. Then, for any positive integer n, all the coefficients of the polynomial (1/n)*Sum_{k=0..n-1} D_k(x)*s_{k+1}(x) are integral and the polynomial is irreducible over the field of rational numbers.
The conjecture was essentially proved by the author in arXiv:1602.00574, except for the irreducibility of (Sum_{k=0..n-1} D_k(x)*s_{k+1}(x))/n. - Zhi-Wei Sun, Feb 01 2016
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LINKS
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FORMULA
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EXAMPLE
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MAPLE
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# Alternative (which also gives an integer for n = 0):
f := n -> hypergeom([-n, -n], [1], 2): # A001850
h := n -> hypergeom([-n, n], [1], 2): # A182626
g := n -> hypergeom([-n, n, 1/2], [1, 1], -8): # A358388
a := n -> (f(n)*((3*n + 1)*f(n) - (-1)^n*(6*n + 3)*h(n)) - n*g(n))/(2*n + 2):
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MATHEMATICA
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d[n_]:=Sum[Binomial[n, k]Binomial[n+k, k], {k, 0, n}]
s[n_]:=Sum[Binomial[n, k]Binomial[n, k-1]/n*2^(k-1), {k, 1, n}]
a[n_]:=Sum[d[k]s[k+1], {k, 0, n-1}]/n
Table[a[n], {n, 1, 20}]
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CROSSREFS
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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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