Richard V. Scholtz, III - cp's OEIS frontend

A228910 a(n) = 8^n - 77^n + 216^n - 355^n + 354^n - 213^n + 72^n - 1.

0, 0, 0, 0, 0, 0, 0, 5040, 181440, 3780000, 59875200, 801496080, 9574044480, 105398092800, 1091804313600, 10794490827120, 102896614941120, 952741767650400, 8617145057539200, 76461500619902160, 667855517349303360, 5757691363157764800, 49099453300298016000, 414884142077935345200

Offset: 0

Richard V. Scholtz, III, Sep 07 2013

Calculates the eighth column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y=1/[1+exp(-x)].

Seiichi Manyama, Table of n, a(n) for n = 0..1107
Index entries for linear recurrences with constant coefficients, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

Cf. A008277, A049434.
The eighth column of results of A163626.
Cf. A000225, A028243, A028244, A028245, A032180, A228909.

[8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*2^(n)-1: n in [0..30]]; // G. C. Greubel, Nov 19 2017

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[ Derivative[n][y][x], y[x]] // Rest; Join[{0, 0, 0, 0, 0, 0, 0}, Table[ -row[n], {n, 7, 23}] [[All, 8]]] (* Jean-François Alcover, Dec 16 2014 *)
Table[7!*StirlingS2[n + 1, 8], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[5040*x^7 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 after Colin Barker *)

a(n)=8^(n)-7*7^(n)+21*6^(n)-35*5^(n)+35*4^(n)-21*3^(n)+7*2^(n)-1.

for(n=0,30, print1(5040*stirling(n+1,8,2), ", ")) \\ G. C. Greubel, Nov 19 2017

a(n) = 5040 * S2(n+1,8), n>=0.
G.f.: 5040*x^7 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Sep 20 2013
E.g.f.: Sum_{k=1..8} (-1)^(8-k)*binomial(8-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Offset corrected by Vaclav Kotesovec, Dec 16 2014

A228909 a(n) = 7^n - 66^n + 155^n - 204^n + 153^n - 6*2^n + 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 20160, 332640, 4233600, 46070640, 451725120, 4115105280, 35517081600, 294293759760, 2362955474880, 18509835445920, 142172988048000, 1074905737084080, 8023358912869440, 59263889194762560, 433988913576556800, 3155502239364459600, 22807773973299268800

Offset: 0

Richard V. Scholtz, III, Sep 07 2013

Essentially Stirling Numbers of the Second Kind, with an offset index, and multiplied by 720.

Calculates the seventh column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y=1/[1+exp(-x)].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A000771, A008277.

Represents the seventh column of results of A163626.

[7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1: n in [0..30]]; // G. C. Greubel, Nov 19 2017

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[ Derivative[n][y][x], y[x]] // Rest; Join[{0, 0, 0, 0, 0, 0}, Table[row[n], {n, 6, 23}] [[All, 7]]] (* Jean-François Alcover, Dec 16 2014 *)
Table[7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[6!*StirlingS2[n + 1, 7], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)

a(n)=7^(n)-6*6^(n)+15*5^(n)-20*4^(n)+15*3^(n)-6*2^(n)+1

concat([0,0,0,0,0,0], Vec(-720*x^6/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)) + O(x^100))) \\ Colin Barker, Dec 16 2014

a(n) = 720 * S(n+1,7), n>=0.
G.f.: -720*x^6 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Dec 16 2014
E.g.f.: Sum_{k=1..7} (-1)^(7-k)*binomial(7-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Offset corrected by Jean-François Alcover, Dec 16 2014
a(20) corrected by Jean-François Alcover, Dec 16 2014
Formula adapted for new offset by Vaclav Kotesovec, Dec 16 2014

A228913 a(n) = 11^n-1010^n+459^n-1208^n+2107^n-2526^n+2105^n-1204^n+453^n-10*2^n+1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 239500800, 8821612800, 239740300800, 5368729766400, 105006251750400, 1858166876966400, 30449278610150400, 469614684719980800, 6897777008118796800, 97349279409046828800, 1329165939158093836800, 17651395149921751680000

Offset: 0

Richard V. Scholtz, III, Sep 07 2013

Calculates the eleventh column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y=1/[1+exp(-x)].

Seiichi Manyama, Table of n, a(n) for n = 0..960
Index entries for linear recurrences with constant coefficients, signature (66, -1925, 32670, -357423, 2637558, -13339535, 45995730, -105258076, 150917976, -120543840, 39916800).

Eleventh column of results of A163626.

Cf. A228910 (with more cf.s), A228911, A228912.

Table[10!*StirlingS2[n+1, 11], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[-3628800*x^10 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)*(11*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
LinearRecurrence[{66,-1925,32670,-357423,2637558,-13339535,45995730,-105258076,150917976,-120543840,39916800},{0,0,0,0,0,0,0,0,0,0,3628800},30] (* Harvey P. Dale, Mar 20 2017 *)

a(n)=11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1

G.f.: -3628800*x^10 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)*(11*x-1)). - Colin Barker, Sep 20 2013

E.g.f.: Sum_{k=1..11} (-1)^(11-k)*binomial(11-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Offset corrected by Vaclav Kotesovec, Dec 16 2014

A228912 a(n) = 10^n - 99^n + 368^n - 847^n + 1266^n - 1265^n + 844^n - 363^n + 92^n - 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 362880, 19958400, 618710400, 14270256000, 273158645760, 4595022432000, 70309810771200, 1000944296352000, 13467262000832640, 173201547619900800, 2147373231974006400, 25832386565857872000, 303056981918271947520, 3481253462769108364800

Offset: 0

Richard V. Scholtz, III, Sep 07 2013

Calculates the tenth column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y = 1/[1+exp(-x)].

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Tenth column of results of A163626.
Essentially 362880*A049435.
Cf. A228910 (with more crossrefs), A228911.

Table[9!*StirlingS2[n+1, 10], {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[10^n-9*9^n+36*8^n-84*7^n+126*6^n-126*5^n+84*4^n-36*3^n+9*2^n-1, {n, 0, 20}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[362880*x^9 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 after Colin Barker *)

a(n)=10^n-9*9^n+36*8^n-84*7^n+126*6^n-126*5^n+84*4^n-36*3^n+9*2^n-1

G.f.: 362880*x^9 / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)*(10*x-1)). - Colin Barker, Sep 20 2013

E.g.f.: Sum_{k=1..10} (-1)^(10-k)*binomial(10-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Offset corrected by Vaclav Kotesovec, Dec 16 2014

A228911 a(n) = 9^n - 88^n + 287^n - 566^n + 705^n - 564^n + 283^n - 8*2^n + 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 40320, 1814400, 46569600, 898128000, 14495120640, 207048441600, 2706620716800, 33094020960000, 384202115256960, 4280991956841600, 46150861752777600, 484294916235312000, 4970346251077025280, 50075960398487654400, 496745174491651008000

Offset: 0

Richard V. Scholtz, III, Sep 07 2013

Calculates the ninth column of coefficients with respect to the derivatives, d^n/dx^n(y), of the logistic equation when written as y=1/[1+exp(-x)].

Essentially 40320 * A049447. - Joerg Arndt, Sep 24 2016

Seiichi Manyama, Table of n, a(n) for n = 0..1047
Index entries for linear recurrences with constant coefficients, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).

The ninth column of results of A163626.

Cf. A228910 (also for more crossrefs).

[9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1: n in [0..32]]; // Vincenzo Librandi, Oct 11 2017

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[ Derivative[n][y][x], y[x]] // Rest; Join[{0, 0, 0, 0, 0, 0, 0, 0}, Table[row[n], {n, 8, 22}] [[All, 9]]] (* Jean-François Alcover, Dec 16 2014 *)
Table[8!*StirlingS2[n + 1, 9], {n, 0, 22}] (* Vaclav Kotesovec, Dec 16 2014 *)
Table[9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1, {n, 0, 22}] (* Vaclav Kotesovec, Dec 16 2014 *)
CoefficientList[Series[-40320*x^8/ ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 16 2014 after Colin Barker *)
lst={}; Do[f= 40320 StirlingS2[n, 9];  AppendTo[lst, f], {n, 1, 5!}]; lst (* Vincenzo Librandi, Oct 11 2017 *)

a(n)=9^n-8*8^n+28*7^n-56*6^n+70*5^n-56*4^n+28*3^n-8*2^n+1

G.f.: -40320*x^8/ ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)*(9*x-1)). - Colin Barker, Sep 20 2013

E.g.f.: Sum_{k=1..9} (-1)^(9-k)*binomial(9-1,k-1)*exp(k*x). - Wolfdieter Lang, May 03 2017

Offset corrected by Vaclav Kotesovec, Dec 16 2014

A163626 Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -7, 12, -6, 1, -15, 50, -60, 24, 1, -31, 180, -390, 360, -120, 1, -63, 602, -2100, 3360, -2520, 720, 1, -127, 1932, -10206, 25200, -31920, 20160, -5040, 1, -255, 6050, -46620, 166824, -317520, 332640, -181440, 40320, 1, -511, 18660

Offset: 0

Richard V. Scholtz, III, Aug 01 2009

Apart from signs and offset, same as A028246. - Joerg Arndt, Nov 06 2016
Triangle T(n,k), read by rows, given by (1,0,2,0,3,0,4,0,5,0,6,0,7,0,8,0,9,...) DELTA (-1,-1,-2,-2,-3,-3,-4,-4,-5,-5,-6,-6,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 05 2011
The "Stirling-Bernoulli transform" maps a sequence b_0, b_1, b_2, ... to a sequence c_0, c_1, c_2, ..., where if B has o.g.f. B(x), c has e.g.f. exp(x)*B(1 - exp(x)). More explicity, c_n = Sum_{k = 0..n} A163626(n,k)*b_k. - Philippe Deléham, May 26 2015
Row sums of absolute values of terms give A000629. - Yahia DJEMMADA, Aug 16 2016
This is the triangle of connection constants for expressing the monomial polynomials (-x)^n as a linear combination of the basis polynomials {binomial(x+n,n)}n>=0, that is, (-x)^n = Sum_{k = 0..n} T(n,k)*binomial(x+k,k). Cf. A145901. - Peter Bala, Jun 06 2019
Row sums for n > 0 are zero. - Shel Kaphan, May 14 2024
The Akiyama-Tanigawa algorithm applied to a sequence yields the same result as the Stirling-Bernoulli Transform applied to the same sequence.  See Philippe Deléham's comment of May 26 2015. - Shel Kaphan, May 16 2024

			y = 1/(1+exp(-x))
y^(0) = y
y^(1) = y-y^2
y^(2) = y-3*y^2+2*y^3
y^(3) = y-7*y^2+12*y^3-6*y^4
Triangle begins :
n\k 0     1     2     3     4     5    6
----------------------------------------
0:  1
1:  1    -1
2:  1    -3     2
3:  1    -7    12    -6
4:  1   -15    50   -60    24
5:  1   -31   180  -390   360  -120
6:  1   -63   602 -2100  3360 -2520  720
7:  1  -127 ... - Reformatted by _Philippe Deléham_, May 26 2015
Change of basis constants: x^4 = 1 - 15*binomial(x+1,1) + 50*binomial(x+2,2) - 60*binomial(x+3,3) + 24*binomial(x+4,4). - _Peter Bala_, Jun 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Logistic function

Cf. A000629, A027642, A028246, A084938, A163626, A164555.
Columns k=0-10 give: A000012, A000225, A028243, A028244, A028245, A032180, A228909, A228910, A228911, A228912, A228913.
Cf. A278075.

A163626 := (n, k) -> add((-1)^j*binomial(k, j)*(j+1)^n, j = 0..k):
for n from 0 to 6 do seq(A163626(n, k), k = 0..n) od; # Peter Luschny, Sep 21 2017

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1-y[x]);
Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n-1][y][x], x];
row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest;
Table[row[n], {n, 0, 9}] // Flatten
(* or *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 9}, {k, 0, n}] // Flatten
(* Jean-François Alcover, Dec 16 2014 *)

from sympy.core.cache import cacheit
@cacheit
def T(n, k):return 1 if n==0 and k==0 else 0 if k>n or k<0 else (k + 1)*T(n - 1, k) - k*T(n - 1, k - 1)
for n in range(51): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Sep 11 2017

T(n, k) = (-1)^k*k!*Stirling2(n+1, k+1). - Jean-François Alcover, Dec 16 2014
T(n, k) = (k+1)*T(n-1,k) - k*T(n-1,k-1), T(0,0) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, May 29 2015
Worpitzky's representation of the Bernoulli numbers B(n, 1) = Sum_{k = 0..n} T(n,k)/(k+1) = A164555(n)/A027642(n) (Bernoulli numbers). - Philippe Deléham, May 29 2015
T(n, k) = Sum_{j=0..k} (-1)^j*binomial(k, j)*(j+1)^n. - Peter Luschny, Sep 21 2017
Let W_n(x) be the row polynomials of this sequence and F_n(x) the row polynomials of A278075. Then W_n(1 - x) = F_n(x). Also Integral_{x=0..1} U_n(x) = Bernoulli(n, 1) for U in {W, F}. - Peter Luschny, Aug 10 2021

A145448 a(n) = 12^n*n!.

Original entry on oeis.org

1, 12, 288, 10368, 497664, 29859840, 2149908480, 180592312320, 17336861982720, 1872381094133760, 224685731296051200, 29658516531078758400, 4270826380475341209600, 666248915354153228697600

Offset: 0

Richard V. Scholtz, III, Oct 10 2008

12-factorial numbers.

Let G(z) = Gamma(z)/(sqrt(2*Pi)*z^(z-1/2)*exp(-z)). For any z > 0 the bounds 1 < G(z) < exp(1/(12*z)) = 1 + 1/(12*z) + 1/(288*z^2) + 1/(10368*z^3) + ... hold. G. Nemes improved the upper bound to 1 + 1/(12*z) + 1/(288*z^2) which gives a simple estimate for the Gamma function on the positive real line. - Peter Luschny, Sep 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Gergő Nemes, Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal, Proceedings of the Royal Society of Edinburgh, 145A, pp. 571-596, 2015.

Cf. A000142, A000165, A032031.

[(Factorial(n)*12^n): n in [0..20]]; // Vincenzo Librandi, Oct 28 2011

Table[12^n*n!, {n,0,30}] (* G. C. Greubel, Mar 24 2022 *)

[12^n*factorial(n) for n in (0..30)] # G. C. Greubel, Mar 24 2022

E.g.f.: 1/(1-12*x). - Philippe Deléham, Oct 28 2011
G.f.: 1/(1 - 12*x/(1 - 12*x/(1 - 24*x/(1 - 24*x/(1 - 36*x/(1 - 36*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Aug 09 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/12).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/12). (End)

a(0)=1 prepended by Richard V. Scholtz, III, Mar 11 2009

a(10)-a(13) corrected by Vincenzo Librandi, Oct 28 2011

cp's OEIS Frontend

User: Richard V. Scholtz, III

A228910 a(n) = 8^n - 7*7^n + 21*6^n - 35*5^n + 35*4^n - 21*3^n + 7*2^n - 1.

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A228909 a(n) = 7^n - 6*6^n + 15*5^n - 20*4^n + 15*3^n - 6*2^n + 1.

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A228913 a(n) = 11^n-10*10^n+45*9^n-120*8^n+210*7^n-252*6^n+210*5^n-120*4^n+45*3^n-10*2^n+1.

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A228912 a(n) = 10^n - 9*9^n + 36*8^n - 84*7^n + 126*6^n - 126*5^n + 84*4^n - 36*3^n + 9*2^n - 1.

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A228911 a(n) = 9^n - 8*8^n + 28*7^n - 56*6^n + 70*5^n - 56*4^n + 28*3^n - 8*2^n + 1.

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A163626 Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).

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A228910 a(n) = 8^n - 77^n + 216^n - 355^n + 354^n - 213^n + 72^n - 1.

A228909 a(n) = 7^n - 66^n + 155^n - 204^n + 153^n - 6*2^n + 1.

A228913 a(n) = 11^n-1010^n+459^n-1208^n+2107^n-2526^n+2105^n-1204^n+453^n-10*2^n+1.

A228912 a(n) = 10^n - 99^n + 368^n - 847^n + 1266^n - 1265^n + 844^n - 363^n + 92^n - 1.

A228911 a(n) = 9^n - 88^n + 287^n - 566^n + 705^n - 564^n + 283^n - 8*2^n + 1.