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A386678 Triangle of numerators for rational convergents to Taylor series of Gamma(x+1).

%I A386678 #12 Aug 14 2025 22:12:46
%S A386678 1,1,0,1,-1,5,1,-17,1045,-35801,1,-181,104905,-38432557,15859708705,1,
%T A386678 -5197,82178809,-864396960373,9983212589988481,-112929359515545345757,
%U A386678 1,-4129,101866157,-213193733657,15527707142596399,-138932602159504972471,2493923095641600267646643,1
%N A386678 Triangle of numerators for rational convergents to Taylor series of Gamma(x+1).
%C A386678 T(n, k) is the numerator of the k-th coefficient in a degree n polynomial approximation to Gamma(x+1) with rational coefficients.
%C A386678 That is, Gamma(x+1) ~ Sum_{j=0..n} A386678(n, j) * x^j / A386679(n, j) which is exact as lim_{n->oo}.
%H A386678 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CauchyProduct.html">Cauchy Product</a>.
%F A386678 Let A(n, k) be the triangle of coefficients A386675(n, k)/A386676(n, k) (see example).
%F A386678 Then T(0, 0) = 1, and for n>=1, T(n, k) = -numerator(Sum_{j=1..k} A(n, j) * T(n, k-j)). This follows immediately from the Cauchy product applied to (1/f(x)) * f(x) = 1.
%F A386678 T(n, k) is also the k-th coefficient in the Taylor series of 1/(Sum_{j=0..n} A(n, j) * x^j).
%F A386678 Equivalently T(n, k) is the k-th coefficient in the polynomial division 1/(Sum_{j=0..n} A(n, j) * x^j).
%e A386678 Let A(n, k) = A386675(n, k)/A386676(n, k) be the triangle
%e A386678   1;
%e A386678   1, 0;
%e A386678   1, 1/4, -1/4;
%e A386678   1, 17/36, -7/12, 1/9;
%e A386678   1, 181/288, -167/192, 77/288, -5/192;
%e A386678   1, 5197/7200, -613/576, 581/1440, -187/2880, 7/1800;
%e A386678   1, 4129/5400, -60239/51840, 5573/11520, -9877/103680, 1597/172800, -37/103680;
%e A386678 where each successive rows gives better rational approximations to 1/Gamma(x+1). Using the Cauchy product, one can obtain approximations to Gamma(x+1) with this table. For instance, T(3, 2) = -numerator(A(3, 1) * T(3, 1) + A(3, 2) * T(3, 0)) = -numerator(17/36 * (-17/36) + (- 7/12) * 1) = 1045. Doing this for each row yields the full table
%e A386678   1;
%e A386678   1, 0;
%e A386678   1, -1/4, 5/16;
%e A386678   1, -17/36, 1045/1296, -35801/46656;
%e A386678   1, -181/288, 104905/82944, -38432557/23887872, 15859708705/6879707136;
%e A386678 As an example, row 3 gives Gamma(x+1) ~ 1 - 17/36x + 1045/1296x^2 - 35801/46656x^3.
%t A386678 Table[Numerator@CoefficientList[Series[1/Sum[Sum[LaguerreL[i,1](-1)^i StirlingS1[i,k]/i!,{i,0,m}] x^k,{k,0,m}],{x,0,m}],x],{m,0,10}]
%Y A386678 Cf. A386675, A386676, A386677, A386679.
%K A386678 sign,frac,easy,tabl
%O A386678 0,6
%A A386678 _David Ulgenes_, Aug 09 2025