Real-rooted atlas

A039755 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 + 2\,x + x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 2\,x^{2}\right)\,P_{n-2}'(x) + \left(4\right)\,P_{n-1}(x) + \left(2 + 2\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3128121
45858151132
5324432172241953
62044339618404003517716
7143202838419516584080048168909
810996025245621047680836153301442631670564
99161122385280233571210998402694163516824088017044017
108215088238744482682547214992384453381676692072912379299179284932
11787770242523808003196867362069932807482592015687168193519213968057001201950431621
1280343952028094619203956817392291333697612287501603093085604738456844386322511308250143112073197252
13867512787232841595136508622346884196184537620262676016597968448011027177921283694729420972428560115721681161824112105
1498776822720402105388608678623871744619923576064337584971152114623503280248549558403487342144318052020187507327001281580819512280317950436
1511819678123525144478074368939029840812894057779146245705387740416219541459737654954815115290819158144998707652873132259235351316110219221112224133674446730525
Technical details
Node id
n-895abc68918cf429 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-809603ba46d41ddf

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:      12,        8,        1;
   4:      58,       58,       15,        1;
   5:     324,      432,      172,       24,       1;
   6:    2044,     3396,     1840,      400,      35,      1;
   7:   14320,    28384,    19516,     5840,     800,     48,     1;
   8:  109960,   252456,   210476,    80836,   15330,   1442,    63,    1;
   9:  916112,  2385280,  2335712,  1099840,  269416,  35168,  2408,   80,  1;
  10: 8215088, 23874448, 26825472, 14992384, 4533816, 766920, 72912, 3792, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(2-2*x-x^2)*P(n-2,x) + (2*x+2*x^2)*d/dx P(n-2,x) + 4*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-(2 - 2*x - x^2)*P[n - 2, x] + (2*x + 2*x^2)*D[P[n - 2, x], x] + 4*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(2 - 2*x - x^2)*v[i - 2] + (2*x + 2*x^2)*deriv(v[i - 2], x) + 4*v[i - 1] + (2 + 2*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, 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.