Real-rooted atlas

A089231 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-3\,x - 10\,x^{2} - 11\,x^{3} - 4\,x^{4} + 3\,n + 14\,n x + 23\,n x^{2} + 16\,n x^{3} + 4\,n x^{4}\right)\,P_{n-1}(x) + \left(-1 - 7\,x - 19\,x^{2} - 25\,x^{3} - 16\,x^{4} - 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1358
213443794
3733756393611448
45013464894610224436127496
540513528512253021199018267562701619232
6376333959401730835402320052432953631788104420516106896
7394353486791925690077751257151314505151375952137976175919748177474633728

All 15 rows

Technical details
Node id
n-ceee4247cb96ddbf · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-452f0d62820ccb13

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,         5;
  2:       13,        44,         37;
  3:       73,       375,        639,         361;
  4:      501,      3464,       8946,       10224,        4361;
  5:     4051,     35285,     122530,      211990,      182675,        62701;
  6:    37633,    395940,    1730835,     4023200,     5243295,      3631788,      1044205;
  7:   394353,   4867919,   25690077,    75125715,   131450515,    137595213,     79761759,     19748177;
  8:  4596553,  65165840,  403325524,  1423212896,  3131278990,   4397910608,   3850097860,   1920434624,   417787921;
  9: 58941091, 944159589, 6709101156, 27754335084, 73653792786, 130016814606, 152646535524, 114921201996, 50335190019, 9770678101;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3*n+(14*n-3)*x+(23*n-10)*x^2+(16*n-11)*x^3+(4*n-4)*x^4)*P(n-1,x) - (1+7*x+19*x^2+25*x^3+16*x^4+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[((3*n + (14*n - 3)*x + (23*n - 10)*x^2 + (16*n - 11)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 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] = ((3*n + (14*n - 3)*x + (23*n - 10)*x^2 + (16*n - 11)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (1 + 7*x + 19*x^2 + 25*x^3 + 16*x^4 + 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

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.