Real-rooted atlas

A103633 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 32\,x + 58\,x^{2} + 44\,x^{3} + 12\,x^{4} - 5\,n - 27\,n x - 51\,n x^{2} - 41\,n x^{3} - 12\,n x^{4}\right)\,P_{n-1}(x) + \left(2 + 16\,x + 48\,x^{2} + 68\,x^{3} + 46\,x^{4} + 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
22101224
3216423696
4440148240144576
54522686848644322304
68120744244044644320172813824
7814411044672117841771214688518455296

All 15 rows

Technical details
Node id
n-b8674991adb819c5 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b19b142556626fc3

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  1,   3;
  2:  2,  10,   12;
  3:  2,  16,   42,    36;
  4:  4,  40,  148,   240,   144;
  5:  4,  52,  268,   684,   864,    432;
  6:  8, 120,  744,  2440,  4464,   4320,   1728;
  7:  8, 144, 1104,  4672, 11784,  17712,  14688,   5184;
  8: 16, 320, 2784, 13760, 42256,  82560, 100224,  69120,  20736;
  9: 16, 368, 3744, 22112, 83536, 209328, 347904, 369792, 228096, 62208;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(5*n-6+(27*n-32)*x+(51*n-58)*x^2+(41*n-44)*x^3+(12*n-12)*x^4)*P(n-1,x) + (2+16*x+48*x^2+68*x^3+46*x^4+12*x^5)*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[(-(5*n - 6 + (27*n - 32)*x + (51*n - 58)*x^2 + (41*n - 44)*x^3 + (12*n - 12)*x^4)*P[n - 1, x] + (2 + 16*x + 48*x^2 + 68*x^3 + 46*x^4 + 12*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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] = (-(5*n - 6 + (27*n - 32)*x + (51*n - 58)*x^2 + (41*n - 44)*x^3 + (12*n - 12)*x^4)*v[i - 1] + (2 + 16*x + 48*x^2 + 68*x^3 + 46*x^4 + 12*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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

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