A174004 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}]]

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, 132, 64, 24, 9, 2, 1, 1, 504, 386, 200, 84, 28, 10, 2, 1, 1, 1309, 1136, 643, 280, 106, 32, 11, 2, 1, 1

Offset

0,4

Comments

Row sums are:

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

Links

Table of n, a(n) for n=0..54.

Formula

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}]]

Example

{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, 132, 64, 24, 9, 2, 1, 1},
{504, 386, 200, 84, 28, 10, 2, 1, 1},
{1309, 1136, 643, 280, 106, 32, 11, 2, 1, 1}

Mathematica

Clear[t, n, q, a];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = f[n - 1, a] + a*f[n - 2, a];
t[n_, q_] = If[n == 0, 1, Sum[Binomial[n, k]*f[k, q], {k, 1, n, 2}]];
a = Table[Table[t[n, q], {n, 0, 10}], {q, 1, 11}];
Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]

Crossrefs

Sequence in context: A179318 A162470 A343234 * A128604 A098885 A106270

Adjacent sequences: A174001 A174002 A174003 * A174005 A174006 A174007

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Mar 05 2010

Status

approved