A154867 A triangular sequence of polynomial coefficients: p(x,n) = Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]); q(x,n)= If[n == 0, 1, p(x, n) + x^n*p(1/x, n)].

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 12, 8, 1, 1, 16, 35, 35, 16, 1, 1, 32, 105, 130, 105, 32, 1, 1, 64, 322, 490, 490, 322, 64, 1, 1, 128, 994, 1967, 2100, 1967, 994, 128, 1, 1, 256, 3061, 8232, 9597, 9597, 8232, 3061, 256, 1, 1, 512, 9375, 34855, 48405, 45654, 48405

Offset

0,5

Comments

Row sums are:

{1, 2, 4, 10, 30, 104, 406, 1754, 8280, 42294, 231950,...}

Links

Table of n, a(n) for n=0..61.

Formula

p(x,n) = Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]);

q(x,n)= If[n == 0, 1, p(x, n) + x^n*p(1/x, n)];

t(n,m)=coefficients(q(x,n)).

Example

{1},
{1, 1},
{1, 2, 1},
{1, 4, 4, 1},
{1, 8, 12, 8, 1},
{1, 16, 35, 35, 16, 1},
{1, 32, 105, 130, 105, 32, 1},
{1, 64, 322, 490, 490, 322, 64, 1},
{1, 128, 994, 1967, 2100, 1967, 994, 128, 1},
{1, 256, 3061, 8232, 9597, 9597, 8232, 3061, 256, 1},
{1, 512, 9375, 34855, 48405, 45654, 48405, 34855, 9375, 512, 1}

Mathematica

Clear[p]; p[x_, n_] = Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]);
q[x_, n_] = If[n == 0, 1, p[x, n] + x^n*p[1/x, n]];
Table[FullSimplify[ExpandAll[q[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[q[x, n]]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A137854 A062715 A100631 * A064298 A256894 A364856

Adjacent sequences: A154864 A154865 A154866 * A154868 A154869 A154870

Keyword

nonn,uned,tabl

Author

Roger L. Bagula, Jan 16 2009

Status

approved