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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

David Ulgenes, Aug 09 2025

T(n, k) is the numerator of the k-th coefficient in a degree n polynomial approximation to Gamma(x+1) with rational coefficients.

That is, Gamma(x+1) ~ Sum_{j=0..n} A386678(n, j) * x^j / A386679(n, j) which is exact as lim_{n->oo}.

			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.

Eric Weisstein's World of Mathematics, Cauchy Product.

Cf. A386675, A386676, A386677, A386679.

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}]

Let A(n, k) be the triangle of coefficients A386675(n, k)/A386676(n, k) (see example).
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).

A386679 Triangle of denominators for rational convergents to Taylor series of Gamma(x+1).

Original entry on oeis.org

1, 1, 1, 1, 4, 16, 1, 36, 1296, 46656, 1, 288, 82944, 23887872, 6879707136, 1, 7200, 51840000, 373248000000, 2687385600000000, 19349176320000000000, 1, 5400, 58320000, 78732000000, 3401222400000000, 18366600960000000000, 198359290368000000000000, 1, 264600, 140026320000, 9262741068000000, 19607370292742400000000

Offset: 0

David Ulgenes, Aug 09 2025

T(n, k) is the denominator of the k-th coefficient in a degree n polynomial approximation to Gamma(x+1) with rational coefficients.

That is, Gamma(x+1) ~ Sum_{j=0..n} A386678(n, j) * x^j / A386679(n, j) which is exact as lim_{n->oo}.

			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) = -denominator(A(3, 1) * T(3, 1) + A(3, 2) * T(3, 0)) = -numerator(17/36 * (-17/36) + (- 7/12) * 1) = 1296. 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.

Eric Weisstein's World of Mathematics, Cauchy Product.

Cf. A386675, A386676, A386677, 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}]

Let A(n, k) be the triangle of coefficients A386675(n, k)/A386676(n, k) (see example).
Then T(0, 0) = 1, and for n>=1, T(n, k) = -denominator(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 denominator of 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 denominator of the k-th coefficient in the polynomial division 1/(Sum_{j=0..n} A(n, j) * x^j).

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

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 36, 12, 9, 1, 288, 192, 288, 192, 1, 7200, 576, 1440, 2880, 1800, 1, 5400, 51840, 11520, 103680, 172800, 103680, 1, 264600, 259200, 1209600, 103680, 44800, 3628800, 2116800, 1, 33868800, 58060800, 58060800, 3686400, 29030400, 4300800, 406425600, 232243200

Offset: 0

David Ulgenes, Jul 28 2025

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.

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}.

			The full triangle is
  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;
  ...
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.

Eric W. Weisstein, Newton's Forward Difference Formula, From MathWorld--A Wolfram Resource.

Cf. A386675.

T[n_, k_] := Denominator[Sum[(-1)^j * StirlingS1[j, k] * LaguerreL[j, 1] / j!,{j, 0, n}]]
maxN = 10;
Table[T[n, k], {n, 0, maxN}, {k, 0, n}]

T(n, k) = denominator( sum(j=0, n, (-1)^j * stirling(j, k, 1) * pollaguerre(j,,1)/j!)); \\ Michel Marcus, Aug 02 2025

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.

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

Original entry on oeis.org

1, 1, 0, 1, 1, -1, 1, 17, -7, 1, 1, 181, -167, 77, -5, 1, 5197, -613, 581, -187, 7, 1, 4129, -60239, 5573, -9877, 1597, -37, 1, 203851, -304867, 600941, -10489, 477, -1907, 17, 1, 25440983, -65392379, 25933147, -277639, 91781, 3029, -40199, 887, 1, 655434541, -3777574277, 11384809949, -12459371, -12541363, 531383, -6199573, 505481, -281

Offset: 0

David Ulgenes, Jul 28 2025

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.

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}.

			The full triangle is
  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;
  ...
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.

Eric W. Weisstein, Newton's Forward Difference Formula, From MathWorld--A Wolfram Resource.

Cf. A386676.

T[n_, k_] := Numerator[Sum[(-1)^j * StirlingS1[j, k] * LaguerreL[j, 1] / j!,{j, 0, n}]]
maxN = 10;
Table[T[n, k], {n, 0, maxN}, {k, 0, n}]

T(n, k) = numerator( sum(j=0, n, (-1)^j * stirling(j, k, 1) * pollaguerre(j,,1)/j!)); \\ Michel Marcus, Aug 02 2025

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.

A386677 Triangle of numerators for rational convergents to Taylor series of 1/Gamma(x+1) (not in lowest terms).

Original entry on oeis.org

1, 1, 0, 4, 1, -1, 36, 17, -21, 4, 576, 362, -501, 154, -15, 14400, 10394, -15325, 5810, -935, 56, 518400, 396384, -602390, 250785, -49385, 4791, -185, 25401600, 19569696, -29876966, 12619761, -2569805, 270459, -13349, 204, 1625702400, 1221167184, -1830986612, 726128116, -122438799, 5139736, 1144962

Offset: 0

David Ulgenes, Jul 29 2025

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.
That is, T(n, k) is the unsimplified version of A386675.
The unsimplified denominators equal (n!)^2 = A001044(n).
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}.

			The simplified triangle of coefficients (A386675) is
  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; ...
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))).
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:
  1/1;
  1/1, 0/1;
  4/4, 1/4, -1/4;
  36/36, 17/36, -21/36, 4/36;
  576/576, 362/576, -501/576, 154/576, -15/576;
  14400/14400, 10394/14400, -15325/14400, 5810/14400, -935/14400, 56/14400; ...
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.

Cf. A386675, A386676.

T[n_, k_] := Numerator[(n!)^2*Sum[(-1)^j * StirlingS1[j, k] * LaguerreL[j, 1] / j!,{j, 0, n}]]
maxN = 10;
Table[T[n, k], {n, 0, maxN}, {k, 0, n}]

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.

A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

Original entry on oeis.org

4, 4, 2, 8, 7, 7, 3, 9, 6, 4, 8, 4, 7, 2, 7, 4, 3, 7, 4, 5, 2, 0, 3, 2, 5, 1, 6, 5, 2, 0, 6, 0, 5, 6, 7, 1, 7, 1, 0, 3, 6, 4, 5, 3, 8, 0, 6, 6, 3, 6, 6, 4, 0, 2, 9, 9, 1, 2, 3, 0, 7, 1, 9, 8, 9, 5, 8, 5, 2, 4, 8, 2, 2, 8, 4, 1, 7, 4, 0, 8, 0, 4, 0, 7, 7, 0, 0, 9, 3, 7, 7, 2, 9, 8, 4, 4, 8, 2, 2, 1, 0, 8, 3, 6, 3, 4

Offset: 0

David Ulgenes, Sep 19 2023

Second branch (i.e., the first after the principal branch) of the inverse gamma function Gamma(y) = x at x=2. See for instance Uchiyama.
Since 1 - x = 0.55712260351... (approximately equal to A249649), we can obtain the interesting approximation Gamma(zeta(2) - zeta(3)) ≈ 2.000001... - David Ulgenes, Feb 19 2024
x is the least positive real number where 1+Gamma(1+Gamma(1+Gamma...(x)...)) converges; it converges to 3. - Colin Linzer, Nov 25 2024

			0.4428773964847274374520325165206056717103645380663664...

K. Amenyo Folitse, David J. Jeffrey, and Robert M. Corless, Properties and Computation of the Functional Inverse of Gamma, 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). p. 65.
Mitsuru Uchiyama, The principal inverse of the gamma function, Proc. Amer. Math. Soc. 140 (2012), 1343-1348.

Cf. A030169, A030171.

Digits:= 140:
with(RootFinding):
NextZero(x -> (x - 1)! - 2, 0);

FindRoot[-2 + (-1 + x)! == 0, {x, 0, 1}, WorkingPrecision -> 15]

solve(x=0.1, 1, gamma(x)-2) \\ Michel Marcus, Sep 19 2023

Equals ((((1/2)!/2)!/2)!/2)!/2...
Proof: Since y = y! / x we substitute the expression into itself to obtain an iterative scheme for the inverse gamma function.
Equals (1/(2*Pi))*Integral_{x=-oo..oo} log((2-Gamma(i*x))/(2-Gamma(1+i*x))) dx. Proof: Follows from writing the inverse gamma function using the Lagrange inversion theorem together with Cauchy's formula for differentiation. - David Ulgenes, Feb 11 2024

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User: David Ulgenes

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

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A386679 Triangle of denominators for rational convergents to Taylor series of Gamma(x+1).

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

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

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A386677 Triangle of numerators for rational convergents to Taylor series of 1/Gamma(x+1) (not in lowest terms).

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A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

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