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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)). 31
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Wikipedia, Logistic function FORMULA 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 EXAMPLE 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 MAPLE 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 MATHEMATICA 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 *) PROG (Python) 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 CROSSREFS 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. Sequence in context: A134436 A306226 A186370 * A028246 A082038 A143774 Adjacent sequences: A163623 A163624 A163625 * A163627 A163628 A163629 KEYWORD easy,sign,tabl AUTHOR Richard V. Scholtz, III, Aug 01 2009 STATUS approved

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Last modified December 4 14:29 EST 2023. Contains 367563 sequences. (Running on oeis4.)