Real-rooted atlas

A099093 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-10\,x - 7\,x^{2} + 12\,n + 16\,n x + 7\,n x^{2}\right)\,P_{n-1}(x) + \left(-12\,x - 20\,x^{2} - 7\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
236421290
321639623445891
412963240295211701718829
57776246243045618468550864887480
646656178848279936229608104328249482457866781
7279936125971223872322474064151696855112410999893158588349
816796168678016193155842422224018741456917244027760324754703531685096170
9100776965878656015032563222138272020712153612778300852031160134942762024190133893843160671
106046617639303014411354204161921200768211015756815731781128067949922812326486380888485142345076278354311569
1136279705626000455688374565376160117793282020382092817676418752109483669444803316848146361038429514002435464392192456082777250160
12217678233617051461632605871083521291859850241841917294081850945230081344889992967122530620827299393712738822254413407184801465290007296561820184055561
13130606940161110158991364313656995841014924764160161390270361618324005898241528887540096949538293056439137282816149419129248363706498085999535864601306902276633638126651751489
147836416409671833817088030300810117127798322718720136850677493761732881998131216333669336320116461099434246314257411200259139890483279270880299217531814312026507778444245326725010487977280521522263450
Technical details
Node id
n-835e923f3e04ffe3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a264a61b871c93de

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,        3;
  2:       36,       42,        12;
  3:      216,      396,       234,        45;
  4:     1296,     3240,      2952,      1170,       171;
  5:     7776,    24624,     30456,     18468,      5508,       648;
  6:    46656,   178848,    279936,    229608,    104328,     24948,     2457;
  7:   279936,  1259712,   2387232,   2474064,   1516968,    551124,   109998,     9315;
  8:  1679616,  8678016,  19315584,  24222240,  18741456,   9172440,  2776032,   475470,   35316;
  9: 10077696, 58786560, 150325632, 221382720, 207121536, 127783008, 52031160, 13494276, 2024190, 133893;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (12*n+(16*n-10)*x+(7*n-7)*x^2)*P(n-1,x) - (12*x+20*x^2+7*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + (16*n - 10)*x + (7*n - 7)*x^2)*P[n - 1, x] - (12*x + 20*x^2 + 7*x^3)*D[P[n - 1, x], x])/(2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + (16*n - 10)*x + (7*n - 7)*x^2)*v[i - 1] - (12*x + 20*x^2 + 7*x^3)*deriv(v[i - 1], x))/(2*n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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