Real-rooted atlas

A130534 → D+1 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 + 2\,x - x^{2} - 2\,n x + 2\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2 - x + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
215511
319231750
41146512074274
51201454807243941764
6127280143538295012248413068
7135490357014609334393933218108109584
81447987812451291570803199763458321499041026576
9154123015540119973585690180098033437403371976138945610628640
101651815287102848231848693792720522077220379820763613579214257440120543840
111772585499956187835139519291648831126934052892409964666922044223797921604606401486442880
12190357582940125139312922338934766694760115361687557586404284223261736042165349561984196544448019802759040
1311044823132132238538329939910268393697173619846480817777042663829768660144568120875379763447299619315226029779840283465647360
1411196370203385432632264809745704222090562874669533179499353143119304328443038439480949875683120132512457902410677792430083706439385604339163001600
Technical details
Node id
n-0254a5a8457ca2c9 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9dfcd5dbd8ed9f8f

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: 1,  2;
  2: 1,  5,    5;
  3: 1,  9,   23,    17;
  4: 1, 14,   65,   120,     74;
  5: 1, 20,  145,   480,    724,    394;
  6: 1, 27,  280,  1435,   3829,   5012,    2484;
  7: 1, 35,  490,  3570,  14609,  33439,   39332,   18108;
  8: 1, 44,  798,  7812,  45129, 157080,  319976,  345832,  149904;
  9: 1, 54, 1230, 15540, 119973, 585690, 1800980, 3343740, 3371976, 1389456;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(2*n-2)*x+(n^2-2*n+1)*x^2)*P(n-2,x) + (2+(2*n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 2)*x + (n^2 - 2*n + 1)*x^2)*P[n - 2, x] + (2 + (2*n - 1)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + (2*n - 2)*x + (n^2 - 2*n + 1)*x^2)*v[i - 2] + (2 + (2*n - 1)*x)*v[i - 1]); 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

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