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

OEIS matches ?

• A163626: triangle, 48 terms agree, 9 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
   1: 1;
   2: 1,   -1;
   3: 1,   -3,     2;
   4: 1,   -7,    12,      -6;
   5: 1,  -15,    50,     -60,      24;
   6: 1,  -31,   180,    -390,     360,     -120;
   7: 1,  -63,   602,   -2100,    3360,    -2520,     720;
   8: 1, -127,  1932,  -10206,   25200,   -31920,   20160,    -5040;
   9: 1, -255,  6050,  -46620,  166824,  -317520,  332640,  -181440,   40320;
  10: 1, -511, 18660, -204630, 1020600, -2739240, 4233600, -3780000, 1814400, -362880;

The row polynomials satisfy P(n,x) = (1-x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 - x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 - x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
  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,  1932,  -10206,   25200,   -31920,   20160,    -5040;
  8: 1, -255,  6050,  -46620,  166824,  -317520,  332640,  -181440,   40320;
  9: 1, -511, 18660, -204630, 1020600, -2739240, 4233600, -3780000, 1814400, -362880;

The row polynomials satisfy P(n,x) = (1-x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 - x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 - x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-61dbccc566bad14f
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n(-x) have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-61dbccc566bad14f
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-61dbccc566bad14f

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• theorem_check ma_wang_interval e-61dbccc566bad14f root: theorem check

  Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 2 proof steps

• e-aa85cc5c316d6389 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: The n-th derivative of the logistic fun…
• e-aa85cc5c316d6389 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: The n-th derivative of the logistic fun…

Related sequences ?

No candidate equalities or related sequences.

References

1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741