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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)).

%I A163626 #89 May 27 2024 15:31:22
%S A163626 1,1,-1,1,-3,2,1,-7,12,-6,1,-15,50,-60,24,1,-31,180,-390,360,-120,1,
%T A163626 -63,602,-2100,3360,-2520,720,1,-127,1932,-10206,25200,-31920,20160,
%U A163626 -5040,1,-255,6050,-46620,166824,-317520,332640,-181440,40320,1,-511,18660
%N A163626 Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).
%C A163626 Apart from signs and offset, same as A028246. - _Joerg Arndt_, Nov 06 2016
%C A163626 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
%C A163626 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
%C A163626 Row sums of absolute values of terms give A000629. - _Yahia DJEMMADA_, Aug 16 2016
%C A163626 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
%C A163626 Row sums for n > 0 are zero. - _Shel Kaphan_, May 14 2024
%C A163626 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
%H A163626 Seiichi Manyama, <a href="/A163626/b163626.txt">Table of n, a(n) for n = 0..10000</a>
%H A163626 Wikipedia, <a href="http://en.wikipedia.org/wiki/Logistic_function">Logistic function</a>
%F A163626 T(n, k) = (-1)^k*k!*Stirling2(n+1, k+1). - _Jean-François Alcover_, Dec 16 2014
%F A163626 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
%F A163626 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
%F A163626 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(k, j)*(j+1)^n. - _Peter Luschny_, Sep 21 2017
%F A163626 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
%e A163626 y = 1/(1+exp(-x))
%e A163626 y^(0) = y
%e A163626 y^(1) = y-y^2
%e A163626 y^(2) = y-3*y^2+2*y^3
%e A163626 y^(3) = y-7*y^2+12*y^3-6*y^4
%e A163626 Triangle begins :
%e A163626 n\k 0     1     2     3     4     5    6
%e A163626 ----------------------------------------
%e A163626 0:  1
%e A163626 1:  1    -1
%e A163626 2:  1    -3     2
%e A163626 3:  1    -7    12    -6
%e A163626 4:  1   -15    50   -60    24
%e A163626 5:  1   -31   180  -390   360  -120
%e A163626 6:  1   -63   602 -2100  3360 -2520  720
%e A163626 7:  1  -127 ... - Reformatted by _Philippe Deléham_, May 26 2015
%e A163626 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
%p A163626 A163626 := (n, k) -> add((-1)^j*binomial(k, j)*(j+1)^n, j = 0..k):
%p A163626 for n from 0 to 6 do seq(A163626(n, k), k = 0..n) od; # _Peter Luschny_, Sep 21 2017
%t A163626 Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1-y[x]);
%t A163626 Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n-1][y][x], x];
%t A163626 row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest;
%t A163626 Table[row[n], {n, 0, 9}] // Flatten
%t A163626 (* or *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 9}, {k, 0, n}] // Flatten
%t A163626 (* _Jean-François Alcover_, Dec 16 2014 *)
%o A163626 (Python)
%o A163626 from sympy.core.cache import cacheit
%o A163626 @cacheit
%o A163626 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)
%o A163626 for n in range(51): print([T(n, k) for k in range(n + 1)]) # _Indranil Ghosh_, Sep 11 2017
%Y A163626 Cf. A000629, A027642, A028246, A084938, A163626, A164555.
%Y A163626 Columns k=0-10 give: A000012, A000225, A028243, A028244, A028245, A032180, A228909, A228910, A228911, A228912, A228913.
%Y A163626 Cf. A278075.
%K A163626 easy,sign,tabl
%O A163626 0,5
%A A163626 _Richard V. Scholtz, III_, Aug 01 2009