Real-rooted atlas

A063967 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(18 + 84\,x + 146\,x^{2} + 112\,x^{3} + 32\,x^{4} - 8\,n - 38\,n x - 68\,n x^{2} - 54\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}(x) + \left(6 + 38\,x + 94\,x^{2} + 114\,x^{3} + 68\,x^{4} + 16\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
36222048
4168816096360
54432086810404642736
61201088392870566304224020736
732835601604038392514723664010816157248
88961132861184182976327168349696206848522241192320
924483532822243279795217836962543744225984011432962521609040896
106688108480780096326400087561281561657618515200140702726218240121753668553216
1118272329056266073612720192398130888523913612641363212822195285121024333941765878784519810048
1249920988288887552047725824170723584426556544759538176963796480854048768503298048177512448283852803941498880
1313638429440002907308817367475269894073619962449924148816960632162790470085117445513193472292082380893566976013705625629886713856
14372608870963293802624616299776275588467288566108162104392384037429571840498325642244904650854434685429760166896721924897153024661766144226618269696
1510179842561600029879948821413740801053319078437617593728100585157760204555976576317925474560375787243008332511749120213570119680941244211202547716096031952896001718350184448
Technical details
Node id
n-9e6d672ceef3b9db · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4600f723d5b687a0

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    2,      4;
   3:    6,     22,     20;
   4:   16,     88,    160,      96;
   5:   44,    320,    868,    1040,     464;
   6:  120,   1088,   3928,    7056,    6304,     2240;
   7:  328,   3560,  16040,   38392,   51472,    36640,    10816;
   8:  896,  11328,  61184,  182976,  327168,   349696,   206848,    52224;
   9: 2448,  35328, 222432,  797952, 1783696,  2543744,  2259840,  1143296,  252160;
  10: 6688, 108480, 780096, 3264000, 8756128, 15616576, 18515200, 14070272, 6218240, 1217536;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(8*n-18+(38*n-84)*x+(68*n-146)*x^2+(54*n-112)*x^3+(16*n-32)*x^4)*P(n-1,x) + (6+38*x+94*x^2+114*x^3+68*x^4+16*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 18 + (38*n - 84)*x + (68*n - 146)*x^2 + (54*n - 112)*x^3 + (16*n - 32)*x^4)*P[n - 1, x] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(8*n - 18 + (38*n - 84)*x + (68*n - 146)*x^2 + (54*n - 112)*x^3 + (16*n - 32)*x^4)*v[i - 1] + (6 + 38*x + 94*x^2 + 114*x^3 + 68*x^4 + 16*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); 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)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [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