Real-rooted atlas

A081579 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(16 + 12\,x\right)\,P_{n-2}(x) + \left(12\,x\right)\,P_{n-2}'(x) + \left(5 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
241519101
3285675288271275
420818175616513508117852
51496593963111960422555751243269137
6108121105047118581491071630245916230857294358101
77800451153867529197008241180478270586128692888938218775282414
8563016112536017544271472550326623024155228755255770624753033242465611380365863
940631525135221532365522869209978373935643492163720268258213327301802867157012137851968326747890379
10293240201145172639919534471318919083651651016082733076266939347295149461234831851864120125938395434994359049545686746938
11211630540515536743987513706616977283555248258367383814964447220525267602265981810476869984675672996878085755340194731535274017714711682335268423
121527337024116592907093751953137669008564953096127110884972760980135958230250150022034275411500475509871180861681418854248202702533219963429953498394531441261695260174841
1311022773768517695829767083276513892906680116925832565761519876175383941631659187707834879762805495288080206898240986640344361032442903575401164234230091514380884229250518441002715943236118337882792584
1479551261228118854074165511138959068673906292081109575681739043740535953471768447368290751702252517609338136047784076919276356801745486090702730231100052364954196569891910621060595032230348146999106297575854782969148954828392645001
Technical details
Node id
n-432d323166de2906 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-964de1fc9d18341b

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:        5,          3;
  2:       41,         51,          9;
  3:      285,        675,        288,         27;
  4:     2081,       8175,       6165,       1350,         81;
  5:    14965,      93963,     111960,      42255,       5751,        243;
  6:   108121,    1050471,    1858149,    1071630,     245916,      23085,       729;
  7:   780045,   11538675,   29197008,   24118047,    8270586,    1286928,     88938,     2187;
  8:  5630161,  125360175,  442714725,  503266230,  241552287,   55255770,   6247530,   332424,    6561;
  9: 40631525, 1352215323, 6552286920, 9978373935, 6434921637, 2026825821, 332730180, 28671570, 1213785, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (16+12*x)*P(n-2,x) + 12*x*d/dx P(n-2,x) + (5+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[(16 + 12*x)*P[n - 2, x] + 12*x*D[P[n - 2, x], x] + (5 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (16 + 12*x)*v[i - 2] + 12*x*deriv(v[i - 2], x) + (5 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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