A124642 Antidiagonal sums of A096465.

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

Author

Gerald McGarvey, Dec 21 2006

Keywords

• nonn

Comments

Apparently bisections give A024718 and A006134 and are related to A078478, A100066 and A105848.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006134, A024718, A078478, A100066, A105848.

Programs

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

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 *)

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

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)

Extensions

Offset changed by Reinhard Zumkeller, Jul 12 2012

Terms a(18) onward added by G. C. Greubel, Apr 30 2021