This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386677 #14 Aug 07 2025 16:46:45
%S A386677 1,1,0,4,1,-1,36,17,-21,4,576,362,-501,154,-15,14400,10394,-15325,
%T A386677 5810,-935,56,518400,396384,-602390,250785,-49385,4791,-185,25401600,
%U A386677 19569696,-29876966,12619761,-2569805,270459,-13349,204,1625702400,1221167184,-1830986612,726128116,-122438799,5139736,1144962
%N A386677 Triangle of numerators for rational convergents to Taylor series of 1/Gamma(x+1) (not in lowest terms).
%C A386677 T(n, k) is the unsimplified (i.e., not in lowest terms) numerator of the k-th coefficient in a degree n polynomial approximation to 1/Gamma(x+1) with rational coefficients.
%C A386677 That is, T(n, k) is the unsimplified version of A386675.
%C A386677 The unsimplified denominators equal (n!)^2 = A001044(n).
%C A386677 Therefore, we have 1/Gamma(x+1) ~ Sum_{j=0..n} A386676(n, j) * x^j / A001044(n) which is exact as lim_{n->oo}.
%F A386677 T(n, k) = numerator( (n!)^2 * Sum_{j=0..n} (-1)^j * Stirling1(j, k) * Lag(j, 1)/j! ) where Lag is the Laguerre L polynomials.
%e A386677 The simplified triangle of coefficients (A386675) is
%e A386677 1;
%e A386677 1, 0;
%e A386677 1, 1/4, -1/4;
%e A386677 1, 17/36, -7/12, 1/9;
%e A386677 1, 181/288, -167/192, 77/288, -5/192;
%e A386677 1, 5197/7200, -613/576, 581/1440, -187/2880, 7/1800;
%e A386677 1, 4129/5400, -60239/51840, 5573/11520, -9877/103680, 1597/172800, -37/103680; ...
%e A386677 These coefficients are obtained using Sum_{j=0..n} (-1)^j * Stirling1(j, k) * Lag(j, 1)/j!. Since Lag(n, x) is in general non-integral, we can write Sum_{j=0..n} (-1)^j * Stirling1(j, k) * numerator(Lag(j, 1))/(j! * denominator(Lag(j, 1))).
%e A386677 Empirically we have LCM(j! * denominator(Lag(j, 1)), {j=0..n}) = (n!)^2. Rescaling so that A001044(n)=(n!)^2 is the denominator of the n-th row gives the following table of coefficients:
%e A386677 1/1;
%e A386677 1/1, 0/1;
%e A386677 4/4, 1/4, -1/4;
%e A386677 36/36, 17/36, -21/36, 4/36;
%e A386677 576/576, 362/576, -501/576, 154/576, -15/576;
%e A386677 14400/14400, 10394/14400, -15325/14400, 5810/14400, -935/14400, 56/14400; ...
%e A386677 Thus for example 36/36 + 17/36x -21/36x^2 + 4/36x^3 is a degree 3 approximation to 1/Gamma(x+1). Therefore, T(3, 1) = 17, T(3, 2) = -21, etc.
%t A386677 T[n_, k_] := Numerator[(n!)^2*Sum[(-1)^j * StirlingS1[j, k] * LaguerreL[j, 1] / j!,{j, 0, n}]]
%t A386677 maxN = 10;
%t A386677 Table[T[n, k], {n, 0, maxN}, {k, 0, n}]
%Y A386677 Cf. A386675, A386676.
%K A386677 sign,frac,easy,tabl
%O A386677 0,4
%A A386677 _David Ulgenes_, Jul 29 2025