A173006 A product triangle sequence based on recursion:a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a)

1, 1, 1, 1, 11, 1, 1, 120, 120, 1, 1, 1309, 14280, 1309, 1, 1, 14279, 1699201, 1699201, 14279, 1, 1, 155760, 202190640, 2205562898, 202190640, 155760, 1, 1, 1699081, 24058986960, 2862818956682, 2862818956682, 24058986960, 1699081, 1, 1

Offset

0,5

Comments

Row sums are:

{1, 2, 13, 242, 16900, 3426962, 2610255700, 5773759285448, 47972252879976100,

1157507562695117906888, 104909162208463229766370000,...}.

a = 1; A034801.

a = 2; A156600.

a = 3; A156602.

This result seems to connect these new recursions directly to q-forms.

Links

Table of n, a(n) for n=0..36.

Formula

a=5; f(n,a)=(2*a+1)*f(n-1,a)+f(n-2,a);

c(n)=If[n == 0, 1, Product[f(i, a), {i, 1, n}]];

t(n,m)=c(n)/(c(m)*c(n-m)

Example

{1},
{1, 1},
{1, 11, 1},
{1, 120, 120, 1},
{1, 1309, 14280, 1309, 1},
{1, 14279, 1699201, 1699201, 14279, 1},
{1, 155760, 202190640, 2205562898, 202190640, 155760, 1},
{1, 1699081, 24058986960, 2862818956682, 2862818956682, 24058986960, 1699081, 1},
{1, 18534131, 2862817257601, 3715936800366098, 40534653607660438, 3715936800366098, 2862817257601, 18534131, 1},
{1, 202176360, 340651194667560, 4823283104057937603, 573930157592104171920, 573930157592104171920, 4823283104057937603, 340651194667560, 202176360, 1},
{1, 2205405829, 40534629348182040, 6260617753130421176727, 8126277060814812179812443, 88644086770258081457215920, 8126277060814812179812443, 6260617753130421176727, 40534629348182040, 2205405829, 1}

Mathematica

Clear[f, c, a, t];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = (2*a + 1)*f[n - 1, a] - f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
t[n_, m_, a_] := c[n, a]/(c[m, a]*c[n - m, a]);
Table[Flatten[Table[Table[t[n, m, a], {m, 0, n}], {n, 0, 10}]], {a, 1, 10}]

Crossrefs

A034801, A156600., A156602.

Sequence in context: A342889 A166979 A022174 * A015125 A290552 A159861

Adjacent sequences: A173003 A173004 A173005 * A173007 A173008 A173009

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Feb 07 2010

Status

approved