Real-rooted atlas

A144084 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 2\,x - 2\,x^{3} - x^{4} + 2\,n x + 5\,n x^{2} + 4\,n x^{3} + n x^{4}\right)\,P_{n-1}(x) + \left(-x^{2} - 3\,x^{3} - 3\,x^{4} - x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
216714
3112393486
4120126292209648
51303101360250515465752
6142645458015675243061332758576
715611971253068635194796263431130922671568

All 15 rows

Technical details
Node id
n-148a4e95b74cc241 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ced16c5d17801fb2

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,  6,    7;
  3: 1, 12,   39,    34;
  4: 1, 20,  126,   292,    209;
  5: 1, 30,  310,  1360,   2505,    1546;
  6: 1, 42,  645,  4580,  15675,   24306,    13327;
  7: 1, 56, 1197, 12530,  68635,  194796,   263431,   130922;
  8: 1, 72, 2044, 29624, 237510, 1068536,  2612092,  3154824,  1441729;
  9: 1, 90, 3276, 62832, 694134, 4541796, 17453100, 37705536, 41368977, 17572114;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(2*n+2)*x+5*n*x^2+(4*n-2)*x^3+(n-1)*x^4)*P(n-1,x) - (x^2+3*x^3+3*x^4+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[((1 + (2*n + 2)*x + 5*n*x^2 + (4*n - 2)*x^3 + (n - 1)*x^4)*P[n - 1, x] - (x^2 + 3*x^3 + 3*x^4 + 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] = ((1 + (2*n + 2)*x + 5*n*x^2 + (4*n - 2)*x^3 + (n - 1)*x^4)*v[i - 1] - (x^2 + 3*x^3 + 3*x^4 + 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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.