Real-rooted atlas

A063967 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(30 + 131\,x + 214\,x^{2} + 155\,x^{3} + 42\,x^{4} - 14\,n - 62\,n x - 103\,n x^{2} - 76\,n x^{3} - 21\,n x^{4}\right)\,P_{n-1}(x) + \left(12 + 68\,x + 153\,x^{2} + 171\,x^{3} + 95\,x^{4} + 21\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2235
36171235
416689645225
5442485234891711475
61208442370332123226489625
732827649690180891896310584245762875
889688003699286272120552100926468729315410625
9244827456134568376440657361733758511245203283353162681875
106688843364721761540488322748245030274183794249606986767213389317515625
11182722558881611120600559214677054245721412854003222707585118443333657231507627114396875
12499207687045376000225394086294481612294037217136165617045545511857849254941814152594281924560747140625
131363842290304176146088204126425772328457526152093551635911168276319713700366002779862501787066313723272965614879671875
14372608677676856845408291191280101628503225519159404743076898660674464568967421925328808533296215477211217822842594059022766336331869765625
15101798419933888181111008101194526438846406161083795022822662411786360797416014394714562340755732687283265925931430803083649649740561059438933104879772208145546875
Technical details
Node id
n-e614fa818116ac77 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-03c8a53a0d7dfacd

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,     3;
   3:    6,    17,     12;
   4:   16,    68,     96,      45;
   5:   44,   248,    523,     489,     171;
   6:  120,   844,   2370,    3321,    2322,     648;
   7:  328,  2764,   9690,   18089,   18963,   10584,    2457;
   8:  896,  8800,  36992,   86272,  120552,  100926,   46872,    9315;
   9: 2448, 27456, 134568,  376440,  657361,  733758,  511245,  203283,  35316;
  10: 6688, 84336, 472176, 1540488, 3227482, 4503027, 4183794, 2496069, 867672, 133893;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(14*n-30+(62*n-131)*x+(103*n-214)*x^2+(76*n-155)*x^3+(21*n-42)*x^4)*P(n-1,x) + (12+68*x+153*x^2+171*x^3+95*x^4+21*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[(-(14*n - 30 + (62*n - 131)*x + (103*n - 214)*x^2 + (76*n - 155)*x^3 + (21*n - 42)*x^4)*P[n - 1, x] + (12 + 68*x + 153*x^2 + 171*x^3 + 95*x^4 + 21*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] = (-(14*n - 30 + (62*n - 131)*x + (103*n - 214)*x^2 + (76*n - 155)*x^3 + (21*n - 42)*x^4)*v[i - 1] + (12 + 68*x + 153*x^2 + 171*x^3 + 95*x^4 + 21*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

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