A386678 Triangle of numerators for rational convergents to Taylor series of Gamma(x+1).
1, 1, 0, 1, -1, 5, 1, -17, 1045, -35801, 1, -181, 104905, -38432557, 15859708705, 1, -5197, 82178809, -864396960373, 9983212589988481, -112929359515545345757, 1, -4129, 101866157, -213193733657, 15527707142596399, -138932602159504972471, 2493923095641600267646643, 1
Offset: 0
Examples
Let A(n, k) = A386675(n, k)/A386676(n, k) be the triangle 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; 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 1; 1, 0; 1, -1/4, 5/16; 1, -17/36, 1045/1296, -35801/46656; 1, -181/288, 104905/82944, -38432557/23887872, 15859708705/6879707136; As an example, row 3 gives Gamma(x+1) ~ 1 - 17/36x + 1045/1296x^2 - 35801/46656x^3.
Links
- Eric Weisstein's World of Mathematics, Cauchy Product.
Programs
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Mathematica
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}]
Formula
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.
T(n, k) is also the k-th coefficient in the Taylor series of 1/(Sum_{j=0..n} A(n, j) * x^j).
Equivalently T(n, k) is the k-th coefficient in the polynomial division 1/(Sum_{j=0..n} A(n, j) * x^j).
Comments