A386677 Triangle of numerators for rational convergents to Taylor series of 1/Gamma(x+1) (not in lowest terms).

1, 1, 0, 4, 1, -1, 36, 17, -21, 4, 576, 362, -501, 154, -15, 14400, 10394, -15325, 5810, -935, 56, 518400, 396384, -602390, 250785, -49385, 4791, -185, 25401600, 19569696, -29876966, 12619761, -2569805, 270459, -13349, 204, 1625702400, 1221167184, -1830986612, 726128116, -122438799, 5139736, 1144962

Offset: 0

Author

David Ulgenes, Jul 29 2025

Keywords

• sign
• frac
• easy
• tabl

Comments

T(n, k) is the unsimplified (i.e., not in lowest terms) numerator of the k-th coefficient in a degree n polynomial approximation to 1/Gamma(x+1) with rational coefficients.

That is, T(n, k) is the unsimplified version of A386675.

The unsimplified denominators equal (n!)^2 = A001044(n).

Therefore, we have 1/Gamma(x+1) ~ Sum_{j=0..n} A386676(n, j) * x^j / A001044(n) which is exact as lim_{n->oo}.

Examples

			The simplified triangle of coefficients (A386675) is
  1;
  1, 0;
  1, 1/4, -1/4;
  1, 17/36, -7/12, 1/9;
  1, 181/288, -167/192, 77/288, -5/192;
  1, 5197/7200, -613/576, 581/1440, -187/2880, 7/1800;
  1, 4129/5400, -60239/51840, 5573/11520, -9877/103680, 1597/172800, -37/103680; ...
These coefficients are obtained using Sum_{j=0..n} (-1)^j * Stirling1(j, k) * Lag(j, 1)/j!. Since Lag(n, x) is in general non-integral, we can write Sum_{j=0..n} (-1)^j * Stirling1(j, k) * numerator(Lag(j, 1))/(j! * denominator(Lag(j, 1))).
Empirically we have LCM(j! * denominator(Lag(j, 1)), {j=0..n}) = (n!)^2. Rescaling so that A001044(n)=(n!)^2 is the denominator of the n-th row gives the following table of coefficients:
  1/1;
  1/1, 0/1;
  4/4, 1/4, -1/4;
  36/36, 17/36, -21/36, 4/36;
  576/576, 362/576, -501/576, 154/576, -15/576;
  14400/14400, 10394/14400, -15325/14400, 5810/14400, -935/14400, 56/14400; ...
Thus for example 36/36 + 17/36x -21/36x^2 + 4/36x^3 is a degree 3 approximation to 1/Gamma(x+1). Therefore, T(3, 1) = 17, T(3, 2) = -21, etc.
		

Crossrefs

Cf. A386675, A386676.

Programs

Mathematica

T[n_, k_] := Numerator[(n!)^2*Sum[(-1)^j * StirlingS1[j, k] * LaguerreL[j, 1] / j!,{j, 0, n}]]
maxN = 10;
Table[T[n, k], {n, 0, maxN}, {k, 0, n}]

Formula

T(n, k) = numerator( (n!)^2 * Sum_{j=0..n} (-1)^j * Stirling1(j, k) * Lag(j, 1)/j! ) where Lag is the Laguerre L polynomials.