Real-rooted atlas

A156992 → basis A008277 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 6\,x + 5\,x^{2} + x^{3} + 2\,n + 5\,n x + 3\,n x^{2} - n^{2} x - 2\,n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4} + n x + 3\,n x^{2} + 3\,n x^{3} + n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
2247860162
3192103217048883816
4192015480415204596018120123000
5230402671201043280184752015328804852805199120
632256052869602840544071789760942026406229440016445520278747280
75160960119145600851195520288735552053357068805533637760303375744068572224018451681920

All 15 rows

Technical details
Node id
n-41d756b3ad7bb8d8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-91f3a2ca42629500

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:          4,           6;
  2:         24,          78,            60;
  3:        192,        1032,          1704,           888;
  4:       1920,       15480,         41520,         45960,          18120;
  5:      23040,      267120,       1043280,       1847520,        1532880,         485280;
  6:     322560,     5286960,      28405440,      71789760,       94202640,       62294400,       16445520;
  7:    5160960,   119145600,     851195520,    2887355520,     5335706880,     5533637760,     3033757440,      685722240;
  8:   92897280,  3031136640,   28173277440,  123497136000,   301796772480,   436759827840,   373182888960,   174368557440,    34411184640;
  9: 1857945600, 86289235200, 1028565216000, 5680944460800, 17652381062400, 33452234323200, 39620751609600, 28713177561600, 11670630854400, 2041544736000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2*n+2-(n^2-5*n-6)*x-(2*n^2-3*n-5)*x^2-(n^2-1)*x^3)*P(n-1,x) + ((n+1)*x+(3*n+3)*x^2+(3*n+3)*x^3+(n+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[((2*n + 2 - (n^2 - 5*n - 6)*x - (2*n^2 - 3*n - 5)*x^2 - (n^2 - 1)*x^3)*P[n - 1, x] + ((n + 1)*x + (3*n + 3)*x^2 + (3*n + 3)*x^3 + (n + 1)*x^4)*D[P[n - 1, x], x])/(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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 2 - (n^2 - 5*n - 6)*x - (2*n^2 - 3*n - 5)*x^2 - (n^2 - 1)*x^3)*v[i - 1] + ((n + 1)*x + (3*n + 3)*x^2 + (3*n + 3)*x^3 + (n + 1)*x^4)*deriv(v[i - 1], x))/(1 + x)); 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.