Real-rooted atlas

A292219 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(54 + 269\,x + 471\,x^{2} + 352\,x^{3} + 96\,x^{4} - 24\,n - 160\,n x - 328\,n x^{2} - 272\,n x^{3} - 80\,n x^{4} + 16\,n^{2} x + 48\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(24 + 200\,x + 552\,x^{2} + 696\,x^{3} + 416\,x^{4} + 96\,x^{5} - 32\,n x - 128\,n x^{2} - 192\,n x^{3} - 128\,n x^{4} - 32\,n x^{5}\right)\,P_{n-1}'(x) + \left(16\,x + 80\,x^{2} + 160\,x^{3} + 160\,x^{4} + 80\,x^{5} + 16\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26713
36014081281
48402940340213038485
515120705601224729381626785328753
6332640194040044906405159880294635066935115539261
786486406054048017513496026831376022981530010441875619674097866545993
825945920021189168007355668320140864724001608707100010963969380413156037066440031155667517781

All 15 rows

Technical details
Node id
n-a97ae88764647980 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4ec588d09c94b2a1

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:            1;
   2:            6,             7;
   3:           60,           140,             81;
   4:          840,          2940,           3402,           1303;
   5:        15120,         70560,         122472,          93816,          26785;
   6:       332640,       1940400,        4490640,        5159880,        2946350,         669351;
   7:      8648640,      60540480,      175134960,      268313760,      229815300,      104418756,       19674097;
   8:    259459200,    2118916800,     7355668320,    14086472400,    16087071000,    10963969380,     4131560370,      664400311;
   9:   8821612800,   82335052800,   333456963840,   766304098560,  1093920828000,   994066557120,   561892210320,   180716884592,   25330978113;
  10: 335221286400, 3519823507200, 16291754519040, 43679333617920, 74824184635200, 84992690633760, 64055711976480, 30902587265232, 8663194514646, 1075631907655;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(24*n-54-(16*n^2-160*n+269)*x-(48*n^2-328*n+471)*x^2-(48*n^2-272*n+352)*x^3-(16*n^2-80*n+96)*x^4)*P(n-1,x) + (24-(32*n-200)*x-(128*n-552)*x^2-(192*n-696)*x^3-(128*n-416)*x^4-(32*n-96)*x^5)*d/dx P(n-1,x) + (16*x+80*x^2+160*x^3+160*x^4+80*x^5+16*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n - 54 - (16*n^2 - 160*n + 269)*x - (48*n^2 - 328*n + 471)*x^2 - (48*n^2 - 272*n + 352)*x^3 - (16*n^2 - 80*n + 96)*x^4)*P[n - 1, x] + (24 - (32*n - 200)*x - (128*n - 552)*x^2 - (192*n - 696)*x^3 - (128*n - 416)*x^4 - (32*n - 96)*x^5)*D[P[n - 1, x], x] + (16*x + 80*x^2 + 160*x^3 + 160*x^4 + 80*x^5 + 16*x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(24*n - 54 - (16*n^2 - 160*n + 269)*x - (48*n^2 - 328*n + 471)*x^2 - (48*n^2 - 272*n + 352)*x^3 - (16*n^2 - 80*n + 96)*x^4)*v[i - 1] + (24 - (32*n - 200)*x - (128*n - 552)*x^2 - (192*n - 696)*x^3 - (128*n - 416)*x^4 - (32*n - 96)*x^5)*deriv(v[i - 1], x) + (16*x + 80*x^2 + 160*x^3 + 160*x^4 + 80*x^5 + 16*x^6)*deriv(deriv(v[i - 1], x), x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.