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

%I A386675 #22 Aug 07 2025 17:05:11
%S A386675 1,1,0,1,1,-1,1,17,-7,1,1,181,-167,77,-5,1,5197,-613,581,-187,7,1,
%T A386675 4129,-60239,5573,-9877,1597,-37,1,203851,-304867,600941,-10489,477,
%U A386675 -1907,17,1,25440983,-65392379,25933147,-277639,91781,3029,-40199,887,1,655434541,-3777574277,11384809949,-12459371,-12541363,531383,-6199573,505481,-281
%N A386675 Triangle of numerators for rational convergents to Taylor series of 1/Gamma(x+1).
%C A386675 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.
%C A386675 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}.
%H A386675 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/NewtonsForwardDifferenceFormula.html">Newton's Forward Difference Formula</a>, From MathWorld--A Wolfram Resource.
%F A386675 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.
%e A386675 The full triangle is
%e A386675   1;
%e A386675   1, 0;
%e A386675   1, 1/4, -1/4;
%e A386675   1, 17/36, -7/12, 1/9;
%e A386675   1, 181/288, -167/192, 77/288, -5/192;
%e A386675   1, 5197/7200, -613/576, 581/1440, -187/2880, 7/1800;
%e A386675   1, 4129/5400, -60239/51840, 5573/11520, -9877/103680, 1597/172800, -37/103680;
%e A386675   ...
%e A386675 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.
%t A386675 T[n_, k_] := Numerator[Sum[(-1)^j * StirlingS1[j, k] * LaguerreL[j, 1] / j!,{j, 0, n}]]
%t A386675 maxN = 10;
%t A386675 Table[T[n, k], {n, 0, maxN}, {k, 0, n}]
%o A386675 (PARI) T(n, k) = numerator( sum(j=0, n, (-1)^j * stirling(j, k, 1) * pollaguerre(j,,1)/j!)); \\ _Michel Marcus_, Aug 02 2025
%Y A386675 Cf. A386676.
%K A386675 sign,frac,easy,tabl
%O A386675 0,8
%A A386675 _David Ulgenes_, Jul 28 2025