Real-rooted atlas

A290598 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(5 + x\right)\,P_{n-1}(x) + \left(18 + 6\,x\right)\,P_{n-1}'(x) + \left(9 + 9\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2516
34316160
4521273331828
58113536094256114472
6154001120965275302410851304992
73442915310137685843598440514512017506432
8884911738922930128881951407025528031597231611210962880
9256845840128494326081050084028174704656148680706818561682820816658246656
1083033591453100050806025411747154927867277796794523366451726381477140272522611232967591424
1129572273445713831860095280173549549488537326738552043503895470294422587211960193029277604189532018944421013504
121149985373458251589808839345217834658185524518681307203405246211386069019286802381093446111222845619705408205617653851373671926866944
134846887070944913710329426979265637743622503959469859133867435601446800205950551285232212281672214204801226263564272621538335943448087978456116868738357268480
14220032252718119401340038008444197453193371540519094158548061918347146068846739310123715877408616336967560102378052362365399952916669182870071265722315156959661217585331818030585280172032
1510701533144929846003173620478927099646401050187076091133680732042605794804528565633847224466971516162203195449901643997167592728352113672736329446950017971897157255450352243124181325090520164437616142399337326195621888
Technical details
Node id
n-4c27e7f9b3484192 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-45f3c3a0aad466d7

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:           5,            1;
   3:          43,           16,           1;
   4:         521,          273,          33,          1;
   5:        8113,         5360,         942,         56,         1;
   6:      154001,       120965,       27530,       2410,        85,        1;
   7:     3442915,      3101376,      858435,      98440,      5145,      120,       1;
   8:    88491173,     89229301,    28881951,    4070255,    280315,     9723,     161,     1;
   9:  2568458401,   2849432608,  1050084028,  174704656,  14868070,   681856,   16828,   208,   1;
  10: 83033591453, 100050806025, 41174715492, 7867277796, 794523366, 45172638, 1477140, 27252, 261, 1;
%F Formula
The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + (18+6*x)*d/dx P(n-1,x) + (9+9*x)*d^2/dx^2 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[(5 + x)*P[n - 1, x] + (18 + 6*x)*D[P[n - 1, x], x] + (9 + 9*x)*D[P[n - 1, x], {x, 2}]];
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] = (5 + x)*v[i - 1] + (18 + 6*x)*deriv(v[i - 1], x) + (9 + 9*x)*deriv(deriv(v[i - 1], x), x)); 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

Proof chains ? JSON

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.