Real-rooted atlas

A133087 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(18 + 76\,x + 119\,x^{2} + 82\,x^{3} + 21\,x^{4} - 15\,n - 66\,n x - 108\,n x^{2} - 78\,n x^{3} - 21\,n x^{4}\right)\,P_{n-1}(x) + \left(10 + 59\,x + 138\,x^{2} + 160\,x^{3} + 92\,x^{4} + 21\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
2412925
312527536175
4169621621681625
5483521032151211073244375
664576216043204860291672915625
719219848784216003186028188138512916109375
8256307216128483849072010886481648349926561390625
97681024060672209664465696689472680400431568159651262442734375
1010241536010368041472010886401959552244944020995201180980393660590499765625
1130725017637248016588804924800102332161518652816096320119410205904900175178723619668359375
124096737286082563041280102643202463436843110144554273285196312034642080155889364251528531441244140625
131228823756821196801155686442958080114960384227867904338722560377598672311778720185335128751103281860043521257641708984375
141638434406433546242012774483026944249080832560431872960740352126097171212609717129457287845158520641934445244464104447829696103515625
Technical details
Node id
n-3d4e79b56d6375ff · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6364361fa4555a56

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:   3,     4;
  2:   4,    12,     9;
  3:  12,    52,    75,     36;
  4:  16,    96,   216,    216,     81;
  5:  48,   352,  1032,   1512,   1107,    324;
  6:  64,   576,  2160,   4320,   4860,   2916,    729;
  7: 192,  1984,  8784,  21600,  31860,  28188,  13851,   2916;
  8: 256,  3072, 16128,  48384,  90720, 108864,  81648,  34992,   6561;
  9: 768, 10240, 60672, 209664, 465696, 689472, 680400, 431568, 159651, 26244;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(15*n-18+(66*n-76)*x+(108*n-119)*x^2+(78*n-82)*x^3+(21*n-21)*x^4)*P(n-1,x) + (10+59*x+138*x^2+160*x^3+92*x^4+21*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[(-(15*n - 18 + (66*n - 76)*x + (108*n - 119)*x^2 + (78*n - 82)*x^3 + (21*n - 21)*x^4)*P[n - 1, x] + (10 + 59*x + 138*x^2 + 160*x^3 + 92*x^4 + 21*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] = (-(15*n - 18 + (66*n - 76)*x + (108*n - 119)*x^2 + (78*n - 82)*x^3 + (21*n - 21)*x^4)*v[i - 1] + (10 + 59*x + 138*x^2 + 160*x^3 + 92*x^4 + 21*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.