Real-rooted atlas

A099097 → basis A013609 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-6 + 6\,n\right)\,P_{n}(x) = \left(62 + 36\,x - 4\,n\right)\,P_{n-1}(x) + \left(93 + 116\,x + 36\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
29615
38411036230
478315309962163525
5729918966184727992129654025
668040220644286092185400600487776828000
763425724655323992040344607216727044328644665612690125
8591243326793342520211165609527236281712140754243032640279936194491875
9551146682852797866458572808353064406750172963490130881127520002080857616796162980828750
10513769311299041097477340442681166522968811308029984730601126431460952968706942721405278721007769645684890625
11478926780330962383326900569956681551889666801754593127521359993551367318720742427000749952653557248093722572860466176700177503125
12446447181603173958282961025470215720198750645172825675221641762321284166496149873130604869103426752022298762419247959755264618770534436279705610731087000000
13416170266849322689644920811465702664912246860602512483586949405956837055607174432279078470859521543907290867262267024401921785446364672345503729664405123379202176782336164467192515625
1438794665561213258056656517412626390498320829898221803732848261938743958456081632575513648264302907129631146930410841615072579082137654019086437376139366943815682451097221120263390662656130606940162520663322734375
1536163709805636327014928304614137270944513306835454129783187686294448397861424812596906607616078665199717792005801214824301440327490619518080014083362668390404541521719767041064925857525761716465431347217022437867527836416409638632285803593750

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-2}(x) + \left(9 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 9 + 6\,x\)
Technical details
Node id
n-f96ce4fdbc0e158a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-98f0bdaf5085291b)
Fingerprint
f6-99e64c87039986de

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:         9,          6;
   3:        84,        110,         36;
   4:       783,       1530,        996,         216;
   5:      7299,      18966,      18472,        7992,        1296;
   6:     68040,     220644,     286092,      185400,       60048,       7776;
   7:    634257,    2465532,    3992040,     3446072,     1672704,     432864,      46656;
   8:   5912433,   26793342,   52021116,    56095272,    36281712,   14075424,    3032640,    279936;
   9:  55114668,  285279786,  645857280,   835306440,   675017296,  349013088,  112752000,  20808576,   1679616;
  10: 513769311, 2990410974, 7734044268, 11665229688, 11308029984, 7306011264, 3146095296, 870694272, 140527872, 10077696;
%F Formula
The row polynomials satisfy (6*n-6)*P(n,x) = -(4*n-62-36*x)*P(n-1,x) + (93+116*x+36*x^2)*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[(-(4*n - 62 - 36*x)*P[n - 1, x] + (93 + 116*x + 36*x^2)*D[P[n - 1, x], x])/(6*n - 6)]];
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] = (-(4*n - 62 - 36*x)*v[i - 1] + (93 + 116*x + 36*x^2)*deriv(v[i - 1], x))/(6*n - 6)); 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