A153354 Triangle of coefficients from a polynomial recursion with row sum near =2*5^n: p(x,n)=(x + 1)*(p(x, n - 1) + 2*5^(n - 2)*(x + 5*x^Floor[n/2] + x^(n - 2))).

2, 5, 5, 2, 46, 2, 2, 123, 123, 2, 2, 250, 746, 250, 2, 2, 502, 2496, 2496, 502, 2, 2, 1754, 4248, 11242, 10498, 1754, 2, 2, 8006, 12252, 46740, 52990, 18502, 8006, 2, 2, 39258, 51508, 58992, 255980, 227742, 57758, 39258, 2, 2, 195510, 247016, 110500, 1096222

Offset

0,1

Comments

Row sums are: {2, 10, 50, 250, 1250, 6000, 29500, 146500, 730500, 3648500, 18234500,...}.

The first five rows are computed by hand to row sum 2*5^n.

The recursion is an approximation form.

Links

Table of n, a(n) for n=0..49.

Formula

p(x,n)=(x + 1)*(p(x, n - 1) + 2*5^(n - 2)*(x + 5*x^Floor[n/2] + x^(n - 2))).

Example

{2},
{5, 5},
{2, 46, 2},
{2, 123, 123, 2},
{2, 250, 746, 250, 2},
{2, 502, 2496, 2496, 502, 2},
{2, 1754, 4248, 11242, 10498, 1754, 2},
{2, 8006, 12252, 46740, 52990, 18502, 8006, 2},
{2, 39258, 51508, 58992, 255980, 227742, 57758, 39258, 2},
{2, 195510, 247016, 110500, 1096222, 1264972, 285500, 253266, 195510, 2},
{2, 976762, 1223776, 357516, 1206722, 6267444, 5456722, 538766, 1230026, 976762, 2}

Mathematica

Clear[p, n, m, x]'
p[x, 0] = 2; p[x, 1] = 5*x + 5;
p[x, 2] = 2*x^2 + 46*x + 2; p[x, 3] = 2*x^3 + 123*x^2 + 123*x + 2;
p[x, 4] = 2 + 250*x + 746*x^2 + 250*x^3 + 2*x^4;
p[x_, n_] := p[x, n] = (x + 1)*(p[x, n - 1] + 2*5^(n - 2)*(x + 5*x^Floor[n/ 2] + x^(n - 2)));
Table[ExpandAll[p[x, n]], {n, 0, 10}];
b = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A187340 A153518 A153648 * A153821 A153879 A074250

Adjacent sequences: A153351 A153352 A153353 * A153355 A153356 A153357

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, Dec 24 2008

Status

approved