Real-rooted atlas

A178820 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(18 + 12\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(12\,x + 4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1241640
23606401601160
34320156809600128030880
4453603046403449601075208960811440
543545651645449676800530432010035205734421641984
6391910480080896234200064201277440649420808257536344064593021184
733592320116640768051416064006513991680319119360067780608061931520196608016788495360

All 15 rows

Technical details
Node id
n-469b0008ce00f6df · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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:         24,           16;
  2:        360,          640,           160;
  3:       4320,        15680,          9600,          1280;
  4:      45360,       304640,        344960,        107520,          8960;
  5:     435456,      5164544,       9676800,       5304320,       1003520,         57344;
  6:    3919104,     80080896,     234200064,     201277440,      64942080,       8257536,       344064;
  7:   33592320,   1166407680,    5141606400,    6513991680,    3191193600,     677806080,     61931520,     1966080;
  8:  277136640,  16222863360,  105391165440,  189500129280,  132360560640,   41783132160,   6282608640,   432537600,   10813440;
  9: 2217093120, 217782906880, 2053823692800, 5110089318400, 4893243801600, 2154404904960, 474147717120, 53115617280, 2854748160, 57671680;
%F Formula
The row polynomials satisfy n*P(n,x) = (6*n+18+(4*n+12)*x)*P(n-1,x) + (4*n+12)*x*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[((6*n + 18 + (4*n + 12)*x)*P[n - 1, x] + (4*n + 12)*x*D[P[n - 1, x], x])/(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] = ((6*n + 18 + (4*n + 12)*x)*v[i - 1] + (4*n + 12)*x*deriv(v[i - 1], x))/(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

  • 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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.