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A386676 Triangle of denominators for rational convergents to Taylor series of 1/Gamma(x+1).

%I A386676 #19 Aug 07 2025 17:05:29
%S A386676 1,1,1,1,4,4,1,36,12,9,1,288,192,288,192,1,7200,576,1440,2880,1800,1,
%T A386676 5400,51840,11520,103680,172800,103680,1,264600,259200,1209600,103680,
%U A386676 44800,3628800,2116800,1,33868800,58060800,58060800,3686400,29030400,4300800,406425600,232243200
%N A386676 Triangle of denominators for rational convergents to Taylor series of 1/Gamma(x+1).
%C A386676 T(n, k) is the denominator of the k-th coefficient in a degree n polynomial approximation to 1/Gamma(x+1) with rational coefficients.
%C A386676 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 A386676 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/NewtonsForwardDifferenceFormula.html">Newton's Forward Difference Formula</a>, From MathWorld--A Wolfram Resource.
%F A386676 T(n, k) = denominator( 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 A386676 The full triangle is
%e A386676   1;
%e A386676   1, 0;
%e A386676   1, 1/4, -1/4;
%e A386676   1, 17/36, -7/12, 1/9;
%e A386676   1, 181/288, -167/192, 77/288, -5/192;
%e A386676   1, 5197/7200, -613/576, 581/1440, -187/2880, 7/1800;
%e A386676   1, 4129/5400, -60239/51840, 5573/11520, -9877/103680, 1597/172800, -37/103680;
%e A386676   ...
%e A386676 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) = 36, T(3, 2) = 12, etc.
%t A386676 T[n_, k_] := Denominator[Sum[(-1)^j * StirlingS1[j, k] * LaguerreL[j, 1] / j!,{j, 0, n}]]
%t A386676 maxN = 10;
%t A386676 Table[T[n, k], {n, 0, maxN}, {k, 0, n}]
%o A386676 (PARI) T(n, k) = denominator( sum(j=0, n, (-1)^j * stirling(j, k, 1) * pollaguerre(j,,1)/j!)); \\ _Michel Marcus_, Aug 02 2025
%Y A386676 Cf. A386675.
%K A386676 nonn,frac,easy,tabl
%O A386676 0,5
%A A386676 _David Ulgenes_, Jul 28 2025