Real-rooted atlas

A026729 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(-9 - 21\,x - 10\,x^{2} + 9\,n + 12\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 18\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
399220
4274521396
581189144455464
62437297833879082240
77292673372625659361741310816
8218794771628114499731721153272152224
965613280567068736294787119035455460334252160
10196831115372646273462752770201414264635994591095551217536
1159049373977101039415374611465290918540385479107433191071965895878784
1217714712400293759453652819572367835398245277627598828123949037746349214428385280
135314414074381136993682674919733861321293648491794798078160952414529517482732246156233137056256
141594323132860254906971910646534715167719815019441210664030755123335208008005673726108988213991410779377661766144
1547829694304672117324976641363824565544390073059359459207120135521910115892200052881660129103472246778263943187656103195289600
Technical details
Node id
n-2affeba2a895b532 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5fb0181067fe4434

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:     3,      1;
   3:     9,      9,      2;
   4:    27,     45,     21,      3;
   5:    81,    189,    144,     45,      5;
   6:   243,    729,    783,    387,     90,      8;
   7:   729,   2673,   3726,   2565,    936,    174,    13;
   8:  2187,   9477,  16281,  14499,   7317,   2115,   327,   21;
   9:  6561,  32805,  67068,  73629,  47871,  19035,  4554,  603,   34;
  10: 19683, 111537, 264627, 346275, 277020, 141426, 46359, 9459, 1095, 55;
%F Formula
The row polynomials satisfy (3*n-3)*P(n,x) = (9*n-9+(12*n-21)*x+(5*n-10)*x^2)*P(n-1,x) - (9*x+18*x^2+5*x^3)*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[((9*n - 9 + (12*n - 21)*x + (5*n - 10)*x^2)*P[n - 1, x] - (9*x + 18*x^2 + 5*x^3)*D[P[n - 1, x], x])/(3*n - 3)]];
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] = ((9*n - 9 + (12*n - 21)*x + (5*n - 10)*x^2)*v[i - 1] - (9*x + 18*x^2 + 5*x^3)*deriv(v[i - 1], x))/(3*n - 3)); 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