Real-rooted atlas

A085880 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
31814234
4135185605385
5113424501358252145208
6102063280227720861010504280430
7962284444445386922494805016043561321383492
8938223609051310198188661560919519502762761801842926089206
993822308433425019027151016630614067597530138738601461460743601430533302770
10956987461178845382352135212040395683902174262090607200594918918007497204306306486211716627494
119917870041662009549264895729756958939066806653248316024291903636495460165257643872037539060125970016796274795761656
12104137635422361525914021193845604952224090709442619656745108009130369928682410916147643751646098234627893301842941105173168587866842290949130
131105461052923379316149300219575006418045180055814533656617609140092328014339132481090858693724821805989485282675329697042007835830089008334821217224208012180044672817436
141184422556700486707525089004041672119940001188431605406500160044222187850011393895730217004677811298160001168377729376001829807719740018159759618001130337208004239730300869193007429004987136104392400
15127917636123607051227573322807450224524924040271251246399588004460949777080160038576011701787560192579110689165205890171893091200114521983959240014432323946040011827491595920620603802000199566712803557005202674440144928867609921320
Technical details
Node id
n-ad42851570f8c7d6 · 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 -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:        3,          1;
   3:       18,         14,          2;
   4:      135,        185,         60,          5;
   5:     1134,       2450,       1358,        252,         14;
   6:    10206,      32802,      27720,       8610,       1050,        42;
   7:    96228,     444444,     538692,     249480,      50160,      4356,      132;
   8:   938223,    6090513,   10198188,    6615609,    1951950,    276276,    18018,     429;
   9:  9382230,   84334250,  190271510,  166306140,   67597530,  13873860,  1461460,   74360,   1430;
  10: 95698746, 1178845382, 3521352120, 4039568390, 2174262090, 607200594, 91891800, 7497204, 306306, 4862;
%F Formula
The row polynomials satisfy n*P(n,x) = (12*n-18+(4*n-6)*x)*P(n-1,x) + (4*n-6)*x*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[((12*n - 18 + (4*n - 6)*x)*P[n - 1, x] + (4*n - 6)*x*D[P[n - 1, x], x])/(n)]];
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] = ((12*n - 18 + (4*n - 6)*x)*v[i - 1] + (4*n - 6)*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, v[i])))};
T(10)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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.