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A124642
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Antidiagonal sums of A096465.
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2
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1, 1, 2, 3, 5, 9, 15, 29, 50, 99, 176, 351, 638, 1275, 2354, 4707, 8789, 17577, 33099, 66197, 125477, 250953, 478193, 956385, 1830271, 3660541, 7030571, 14061141, 27088871, 54177741, 104647631, 209295261, 405187826, 810375651, 1571990936, 3143981871, 6109558586, 12219117171, 23782190486, 47564380971, 92705454896
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OFFSET
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0,3
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COMMENTS
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Apparently bisections give A024718 and A006134 and are related to A078478, A100066 and A105848.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
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FORMULA
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Conjecture: G.f.: -(1/2)*z*(2*z+(1-4*z^2)^(1/2)+1)/(1-4*z^2)^(1/2)/(z^2-1). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
From G. C. Greubel, Apr 30 2021: (Start)
a(n) = (1 + (-1)^n)/2 + Sum_{j=0..floor((n-1)/2)} Sum_{k=0..j} (n-2*j)*binomial(n -2*k, n-k-j)/(n-2*k).
a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} ((n-2*j)/(n-k-j))*binomial(n-2*k, n-k-j). (End)
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MATHEMATICA
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a[_, 0]=1; a[n_, n_]=1; a[n_, m_]:= a[n, m] = a[n-1, m] + a[n, m-1]; a[n_, m_] /; n<0 || m>n = 0; Table[ Sum[a[n-m, m], {m, 0, n}], {n, 0, 45}] (* Jean-François Alcover, Dec 17 2012 *)
a[n_]:= a[n]= (1+(-1)^n)/2 + Sum[(n-2*j)*Binomial[n-2*k, n-k-j]/(n-2*k), {j, 0, (n-1)/2}, {k, 0, j}]; Table[a[n], {n, 0, 45}] (* G. C. Greubel, Apr 30 2021 *)
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PROG
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(Magma)
a:= func< n | n eq 0 select 1 else (1+(-1)^n)/2 + (&+[ (&+[ ((n-2*j)/(n-2*k))*Binomial(n-2*k, n-k-j) : k in [0..j]]) : j in [0..Floor((n-1)/2)]]) >;
[a(n): n in [0..45]]; // G. C. Greubel, Apr 30 2021
(Sage)
def a(n): return (1+(-1)^n)/2 + sum( sum( ((n-2*j)/(n-2*k))*binomial(n-2*k, n-k-j) for k in (0..j)) for j in (0..(n-1)//2))
[a(n) for n in (0..45)] # G. C. Greubel, Apr 30 2021
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CROSSREFS
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Cf. A006134, A024718, A078478, A100066, A105848.
Sequence in context: A351359 A191701 A066726 * A269153 A232866 A011826
Adjacent sequences: A124639 A124640 A124641 * A124643 A124644 A124645
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KEYWORD
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nonn
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AUTHOR
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Gerald McGarvey, Dec 21 2006
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EXTENSIONS
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Offset changed by Reinhard Zumkeller, Jul 12 2012
Terms a(18) onward added by G. C. Greubel, Apr 30 2021
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STATUS
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approved
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