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A099576 Row sums of triangle A099575.

%I A099576 #25 Dec 26 2023 11:23:00
%S A099576 1,2,6,12,35,72,210,440,1287,2730,8008,17136,50388,108528,319770,
%T A099576 692208,2042975,4440150,13123110,28614300,84672315,185122080,
%U A099576 548354040,1201610592,3562467300,7821594872,23206929840,51037462560,151532656696
%N A099576 Row sums of triangle A099575.
%H A099576 Andrew Howroyd, <a href="/A099576/b099576.txt">Table of n, a(n) for n = 0..200</a>
%F A099576 a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} binomial(n+j, j).
%F A099576 Conjecture: 4*n*(n-1)*(3*n+2)*(n+2)*a(n) - 36*(n-1)*(n+1)*a(n-1) - 3*n*(3*n+5)*(3*n-1)*(3*n-2)*a(n-2) = 0. - _R. J. Mathar_, Nov 28 2014
%F A099576 From _Robert Israel_, May 08 2018: (Start)
%F A099576 a(2*n) = (1+n/(n+1))*binomial(3*n+1,n).
%F A099576 a(2*n+1) = 2*binomial(3*n+3,n).
%F A099576 The conjecture follows from this. (End)
%F A099576 a(n) = (1/(n+1))*Sum_{k=0..n} binomial(n + floor(k/2) + 1, floor(k/2) + 1)*(1 + floor(k/2)). - _G. C. Greubel_, Jul 24 2022
%F A099576 a(n) = binomial(2*n+2, n)*hypergeom([-n, n+1], [-2*n-2], -1). - _Detlef Meya_, Dec 25 2023
%p A099576 seq(op([(1+n/(n+1))*binomial(3*n+1,n),2*binomial(3*n+3,n)]),n=0..20);
%t A099576 a[n_] := Sum[Binomial[n + j, j], {k, 0, n}, {j, 0, k/2}];
%t A099576 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 06 2018 *)
%t A099576 a[n_] := Binomial[2*n+2, n]*Hypergeometric2F1[-n, n+1, -2*n-2, -1]; Flatten[Table[a[n], {n, 0, 28}]] (* _Detlef Meya_, Dec 25 2023 *)
%o A099576 (PARI) a(n) = sum(k=0, n, sum(j=0, floor(k/2), binomial(n+j, j))); \\ _Andrew Howroyd_, Feb 13 2018
%o A099576 (Magma) [(&+[Binomial(n+Floor(k/2)+1, Floor(k/2)+1)*(1+Floor(k/2))/(n+1): k in [0..n]]): n in [0..40]]; // _G. C. Greubel_, Jul 24 2022
%o A099576 (SageMath) [sum( binomial(n+(k//2)+1, (k//2)+1)*(1+(k//2))/(n+1) for k in (0..n) ) for n in (0..40)] # _G. C. Greubel_, Jul 24 2022
%Y A099576 Cf. A029907, A099575, A099577.
%K A099576 easy,nonn
%O A099576 0,2
%A A099576 _Paul Barry_, Oct 23 2004