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Triangle, read by rows, T(n,k) = Sum_{m=0..(n-k)/2} C(k,m)*C(n-2*m,k).
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%I #20 Jan 21 2019 12:00:45

%S 1,1,1,1,2,1,1,4,3,1,1,6,8,4,1,1,8,16,13,5,1,1,10,28,32,19,6,1,1,12,

%T 44,68,55,26,7,1,1,14,64,128,136,86,34,8,1,1,16,88,220,296,241,126,43,

%U 9,1,1,18,116,352,584,592,393,176,53,10,1

%N Triangle, read by rows, T(n,k) = Sum_{m=0..(n-k)/2} C(k,m)*C(n-2*m,k).

%C From _Emanuele Munarini_, Feb 21 2017: (Start)

%C T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps X=(1,0), D=(1,1) and E=(3,1).

%C Row sums = A008998.

%C Central coefficients = A006139. (End)

%H Michael De Vlieger, <a href="/A257365/b257365.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)

%H James East, Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.

%F G.f.: 1/(1-y-x*(1+y^2)).

%F From _Emanuele Munarini_, Feb 21 2017: (Start)

%F G.f. for the triangle: 1/(1-x-x*y-x^3*y).

%F Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n,k). (End)

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 4, 3, 1;

%e 1, 6, 8, 4, 1;

%e 1, 8, 16, 13, 5, 1;

%t Table[Sum[Binomial[k, m] Binomial[n - 2 m, k], {m, 0, (n - k)/2}], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Apr 21 2015 *)

%o (Maxima) T(n,k):=sum(binomial(k,m)*binomial(n-2*m,k),m,0,(n-k)/2);

%Y Cf. A006139.

%K nonn,tabl

%O 0,5

%A _Vladimir Kruchinin_, Apr 21 2015