Real-rooted atlas

A038338 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(60 + 36\,x\right)\,P_{n-1}(x) + \left(36\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 3600 + 5616\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
036005616129610512
1216000668736373248466561304640
2129600007197465673343232212751361679616181232640
3777600000737612697612272394240621457920011085465606046617627809713152
446656000000736101789696188549661081614858554982404498683494405441955840021767823364660574588928
527993600000007234538781081627538521705676831750160169369614526394122240029255954595840255989602713678364164096845189722570752
616796160000000070459341898383363895528261268275263254136898584576410639016703426561225093098700800017597456689397761166058761748482821109907456164617320613478400
71007769600000000068245629982448025653957509357083033601202908517301878784010654184869098946560441852389701425561692672332015966617699731877448384512517955779008921610155995666841634221815186019581952

All 15 rows

Technical details
Node id
n-ccf209b8477ad005 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c377b8329dc9dd7c

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:                 3600,                   5616,                    1296;
  1:               216000,                 668736,                  373248,                    46656;
  2:             12960000,               71974656,                73343232,                 21275136,                  1679616;
  3:            777600000,             7376126976,             12272394240,               6214579200,               1108546560,                 60466176;
  4:          46656000000,           736101789696,           1885496610816,            1485855498240,             449868349440,              54419558400,               2176782336;
  5:        2799360000000,         72345387810816,         275385217056768,          317501601693696,          145263941222400,           29255954595840,            2559896027136,             78364164096;
  6:      167961600000000,       7045934189838336,       38955282612682752,        63254136898584576,        41063901670342656,        12250930987008000,         1759745668939776,         116605876174848,          2821109907456;
  7:    10077696000000000,     682456299824480256,     5395750935708303360,     12029085173018787840,     10654184869098946560,      4418523897014255616,       926723320159666176,       99731877448384512,       5179557790089216,       101559956668416;
  8:   604661760000000000,   65878601839150104576,   736807550307177332736,   2215133342752655278080,   2606500779524861460480,   1443996390570983424000,    414842496656581066752,    64478385289643950080,    5392833699092889600,    225463103803883520,    3656158440062976;
  9: 36279705600000000000, 6346113599918410039296, 99630226306756811685888, 398667473393504470695936, 611470959362167327948800, 440393161799931034337280, 166480399137771777687552, 35051586090005830828032, 4197927997711507783680, 280719845028035297280, 9652258281766256640, 131621703842267136;
%F Formula
The row polynomials satisfy P(n,x) = (60+36*x)*P(n-1,x) + 36*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3600 + 5616*x + 1296*x^2.
%t Mathematica
P[0, x_] := 3600 + 5616*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[(60 + 36*x)*P[n - 1, x] + 36*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] = 3600 + 5616*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 36*x)*v[i - 1] + 36*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

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.