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A triangle based on:a(n,q)=a(n-1,q)+q*a(n-2,q);t(n,q)=If[n == 0, 1, Sum[Binomial[n, k]*a(k, q), {k, 1, n, 2}]]
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%I #6 May 23 2012 10:29:29

%S 1,1,1,2,1,1,5,2,1,1,12,6,2,1,1,30,16,7,2,1,1,76,46,20,8,2,1,1,195,

%T 132,64,24,9,2,1,1,504,386,200,84,28,10,2,1,1,1309,1136,643,280,106,

%U 32,11,2,1,1

%N A triangle based on:a(n,q)=a(n-1,q)+q*a(n-2,q);t(n,q)=If[n == 0, 1, Sum[Binomial[n, k]*a(k, q), {k, 1, n, 2}]]

%C Row sums are:

%C {1, 2, 4, 9, 22, 57, 154, 428, 1216, 3521,...}.

%F a(n,q)=a(n-1,q)+q*a(n-2,q);

%F t(n,q)=If[n == 0, 1, Sum[Binomial[n, k]*a(k, q), {k, 1, n, 2}]]

%e {1},

%e {1, 1},

%e {2, 1, 1},

%e {5, 2, 1, 1},

%e {12, 6, 2, 1, 1},

%e {30, 16, 7, 2, 1, 1},

%e {76, 46, 20, 8, 2, 1, 1},

%e {195, 132, 64, 24, 9, 2, 1, 1},

%e {504, 386, 200, 84, 28, 10, 2, 1, 1},

%e {1309, 1136, 643, 280, 106, 32, 11, 2, 1, 1}

%t Clear[t, n, q, a];

%t f[0, a_] := 0; f[1, a_] := 1;

%t f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];

%t t[n_, q_] = If[n == 0, 1, Sum[Binomial[n, k]*f[k, q], {k, 1, n, 2}]];

%t a = Table[Table[t[n, q], {n, 0, 10}], {q, 1, 11}];

%t Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];

%t Flatten[%]

%K nonn,tabl,uned

%O 0,4

%A _Roger L. Bagula_, Mar 05 2010