A386675 Triangle of numerators for rational convergents to Taylor series of 1/Gamma(x+1).

1, 1, 0, 1, 1, -1, 1, 17, -7, 1, 1, 181, -167, 77, -5, 1, 5197, -613, 581, -187, 7, 1, 4129, -60239, 5573, -9877, 1597, -37, 1, 203851, -304867, 600941, -10489, 477, -1907, 17, 1, 25440983, -65392379, 25933147, -277639, 91781, 3029, -40199, 887, 1, 655434541, -3777574277, 11384809949, -12459371, -12541363, 531383, -6199573, 505481, -281

Offset: 0

Author

David Ulgenes, Jul 28 2025

Keywords

• sign
• frac
• easy
• tabl

Comments

T(n, k) is the numerator of the k-th coefficient in a degree n polynomial approximation to 1/Gamma(x+1) with rational coefficients.

That is, 1/Gamma(x+1) ~ Sum_{j=0..n} A386675(n, j) * x^j / A386676(n, j) which is exact as lim_{n->oo}.

Examples

			The full triangle 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;
  ...
Thus, for example, a degree 3 approximation is 1/Gamma(x+1) ~ 1 + 17/36x - 7/12x^2 + 1/9x^3. Therefore, T(3, 1) = 17, T(3, 2) = -7, etc.
		

Links

• Eric W. Weisstein, Newton's Forward Difference Formula, From MathWorld--A Wolfram Resource.

Crossrefs

Cf. A386676.

Programs

Mathematica

T[n_, k_] := Numerator[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}]

PARI

T(n, k) = numerator( sum(j=0, n, (-1)^j * stirling(j, k, 1) * pollaguerre(j,,1)/j!)); \\ Michel Marcus, Aug 02 2025

Formula

T(n, k) = numerator( Sum_{j=0..n} (-1)^j * Stirling1(j, k) * Lag(j, 1)/j! ) where Lag(n, x) is the Laguerre polynomial. Proof: Apply Newton's Forward Difference Formula to f(n) = 1/n!. Use the identity x * (x-1) * ... * (x - n + 1) = Sum_{k=0..n} Stirling1(n, k) * x^k and interchange the order of summation.