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A124642
Antidiagonal sums of A096465.
2
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
OFFSET
0,3
COMMENTS
Apparently bisections give A024718 and A006134 and are related to A078478, A100066 and A105848.
LINKS
FORMULA
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)
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gerald McGarvey, Dec 21 2006
EXTENSIONS
Offset changed by Reinhard Zumkeller, Jul 12 2012
Terms a(18) onward added by G. C. Greubel, Apr 30 2021
STATUS
approved