A144452 Antidiagonal expansion of the polynomials: f(x,n) = 1/(exp(t) - Sum_{i=1..n} t^i/i!).

1, 1, 0, 1, 0, -3, 1, 0, 0, -4, 1, 0, 0, -4, 25, 1, 0, 0, 0, -5, 114, 1, 0, 0, 0, -5, -6, -287, 1, 0, 0, 0, 0, -6, 133, -4152, 1, 0, 0, 0, 0, -6, -7, 552, -1647, 1, 0, 0, 0, 0, 0, -7, -8, 1629, 192230, 1, 0, 0, 0, 0, 0, -7, -8, 621, -12610, 807961, 1, 0, 0, 0, 0, 0, 0, -8, -9, 2510, -128579, -10164804, 1, 0, 0, 0, 0, 0, 0, -8, -9, -10, 7381

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Oct 06 2008

Keywords

• uned
• sign

Comments

Row sums are:

{1, 2, 3, 20, 125, 804, 4501, 36896, 362673, 3831560, 40591001, 467518248, 6106124713, 87661533764, 1323052370025}.

Triangle sequence rows last terms are:

Table[n!*a[[1]][[n]], {n, 1, 15}]

{1, 0, -3, -4, 25,114, -287, -4152, -1647, 192230, 807961, -10164804, -111209111, 454840554, 14657978385}

Examples

			{1},
{1, 0},
{1, 0, -3},
{1, 0, 0, -4},
{1, 0, 0, -4, 25},
{1, 0, 0, 0, -5, 114},
{1, 0, 0, 0, -5, -6, -287},
{1, 0, 0, 0, 0, -6, 133, -4152},
{1, 0, 0, 0, 0, -6, -7, 552, -1647},
{1, 0, 0, 0, 0, 0, -7, -8,1629, 192230},
{1, 0, 0, 0, 0, 0, -7, -8, 621, -12610, 807961},
{1, 0, 0, 0, 0, 0, 0, -8, -9, 2510, -128579, -10164804},
{1, 0, 0, 0, 0, 0, 0, -8, -9, -10, 7381, -725484, -111209111},
{1, 0, 0, 0, 0,0, 0, 0, -9, -10, 2761, 18996, 1522651, 454840554},
{1, 0, 0, 0, 0, 0, 0,0, -9, -10, -11, 11076, -404989, 54082014, 14657978385}
		

Crossrefs

Cf. A089148.

Programs

Mathematica

Clear[f, b, a, g, h, n, t]; f[t_, n_] = 1/(Exp[t] - Sum[t^i/i!, {i, 1, n}]); a = Table[Table[SeriesCoefficient[Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}]; b = Table[Table[m!*a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}]; Flatten[b]

Formula

f(x,n) = 1/(exp(t) - Sum_{i=1..n} t^i/i!); t(n,m) = Expansion(f(x,n)); t_out(n,m) = m!*t(n-m+1,m).