Real-rooted atlas

A133087 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 4\,n x - n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(64 + 28\,x - 2\,x^{3} - 40\,n + 30\,n x + 16\,n x^{2} + n x^{3} - 30\,n^{2} x - 3\,n^{2} x^{2} + 3\,n^{2} x^{3} - 5\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(40 + 32\,x + 2\,x^{2} - 4\,x^{3} - x^{4} + 40\,n x + 34\,n x^{2} + 7\,n x^{3} + 10\,n^{2} x^{2} + 9\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
248315
3123221469
4166472325189
548224312192556837
6643847206403007272187
7192128027362880170057610589477
825620485376716856002688784128924057
97686656192002867225760147845488128017110102789
1010241024034560614406720048384235207680162020011255879
1130723276811904023040027840022579212700849920135002400253121082565
1240964915220275245056063360060825641395220275271280176002904288132657205
1312288155648681984162201624640002585088195148810813444419361320002807240323511411160261
141638422937611182082981888512512061501445381376351436817297286406401761763494447323921527103491
Technical details
Node id
n-7cecb752cbb3a1ef · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6f892c34dd4e60b6

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:   3,    2;
  2:   4,    8,     3;
  3:  12,   32,    21,     4;
  4:  16,   64,    72,    32,     5;
  5:  48,  224,   312,   192,    55,     6;
  6:  64,  384,   720,   640,   300,    72,    7;
  7: 192, 1280,  2736,  2880,  1700,   576,  105,    8;
  8: 256, 2048,  5376,  7168,  5600,  2688,  784,  128,   9;
  9: 768, 6656, 19200, 28672, 25760, 14784, 5488, 1280, 171, 10;
%F Formula
The row polynomials satisfy (8+4*n*x+(n^2-n)*x^2)*P(n,x) = -(40*n-64+(30*n^2-30*n-28)*x+(5*n^3+3*n^2-16*n)*x^2+(2*n^3-3*n^2-n+2)*x^3)*P(n-1,x) + (40+(40*n+32)*x+(10*n^2+34*n+2)*x^2+(9*n^2+7*n-4)*x^3+(2*n^2-1)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(40*n - 64 + (30*n^2 - 30*n - 28)*x + (5*n^3 + 3*n^2 - 16*n)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*P[n - 1, x] + (40 + (40*n + 32)*x + (10*n^2 + 34*n + 2)*x^2 + (9*n^2 + 7*n - 4)*x^3 + (2*n^2 - 1)*x^4)*D[P[n - 1, x], x])/(8 + 4*n*x + (n^2 - n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(40*n - 64 + (30*n^2 - 30*n - 28)*x + (5*n^3 + 3*n^2 - 16*n)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*v[i - 1] + (40 + (40*n + 32)*x + (10*n^2 + 34*n + 2)*x^2 + (9*n^2 + 7*n - 4)*x^3 + (2*n^2 - 1)*x^4)*deriv(v[i - 1], x))/(8 + 4*n*x + (n^2 - n)*x^2)); 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

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.