Real-rooted atlas

A178821 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(12 + 4\,x + 3\,n + n x\right)\,P_{n-1}(x) + \left(4\,x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115520
213510515255
39451295420352695
4567012250679012607026040
5306189840683160258303150126241290
61530907070708570103969007980069302102201010
772171046850107844760508893015015002125201386033020068620
8324769529192625658632155756751023399145480249050589025740495184604805
91407334517335961551784590059405417531974442589294205135135001102530450457151723033455
1059108049990516527386762476157150393303965171210144773729129528198734354320223723575075100116377146786
1124180565554834193052772087318052033446465456425970002120054937055981025108711252558040532542792751201201365158876724825
129672226202956731050019211704694045322873596049537550062028699598928095444629280190791400802340778440175675500778778018564018201575300004460
1337944887401559246075801294814866800380733237430051272748527003650218311740149861231520037430865472058620842280581777196036212176013582660278460238015977095071280
14146358851408067766102208524284353460310356117162005104061529840044137313159940220342231909806739327108800131032018260016512956460013532599080710073000228337204069803060165822502986180
Technical details
Node id
n-17e55d1ba7fd6f6e · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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:       15,         5;
  2:      135,       105,        15;
  3:      945,      1295,       420,        35;
  4:     5670,     12250,      6790,      1260,        70;
  5:    30618,     98406,     83160,     25830,      3150,      126;
  6:   153090,    707070,    857010,    396900,     79800,     6930,      210;
  7:   721710,   4685010,   7844760,   5088930,   1501500,   212520,    13860,     330;
  8:  3247695,  29192625,  65863215,  57567510,  23399145,  4802490,   505890,   25740,   495;
  9: 14073345, 173359615, 517845900, 594054175, 319744425, 89294205, 13513500, 1102530, 45045, 715;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n+12+(n+4)*x)*P(n-1,x) + (n+4)*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[((3*n + 12 + (n + 4)*x)*P[n - 1, x] + (n + 4)*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] = ((3*n + 12 + (n + 4)*x)*v[i - 1] + (n + 4)*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.