Real-rooted atlas

A048854 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(18 + 165\,x + 371\,x^{2} + 320\,x^{3} + 96\,x^{4} - 8\,n - 112\,n x - 280\,n x^{2} - 256\,n x^{3} - 80\,n x^{4} + 16\,n^{2} x + 48\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(8 + 136\,x + 456\,x^{2} + 632\,x^{3} + 400\,x^{4} + 96\,x^{5} - 32\,n x - 128\,n x^{2} - 192\,n x^{3} - 128\,n x^{4} - 32\,n x^{5}\right)\,P_{n-1}'(x) + \left(16\,x + 80\,x^{2} + 160\,x^{3} + 160\,x^{4} + 80\,x^{5} + 16\,x^{6}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
312362573
41205407503311741
51680100802100018536593757233
6302402268006300008341205343301336512389141
76652805987520207900003670128035265780176419323609673120661465
8172972801816214407567560001669908240213945732016054158126569604861135255957140942173

All 15 rows

Technical details
Node id
n-bdea7124c5a6f3dd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-7d14cc12da244347

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:           2,            3;
   3:          12,           36,            25;
   4:         120,          540,           750,           331;
   5:        1680,        10080,         21000,         18536,          5937;
   6:       30240,       226800,        630000,        834120,        534330,        133651;
   7:      665280,      5987520,      20790000,      36701280,      35265780,      17641932,       3609673;
   8:    17297280,    181621440,     756756000,    1669908240,    2139457320,    1605415812,     656960486,     113525595;
   9:   518918400,   6227020800,   30270240000,   80155595520,  128367439200,  128433264960,   78835258320,   27246142800,    4070168161;
  10: 17643225600, 238183545600, 1323241920000, 4087935371520, 7856087279040, 9825144769440, 8041196348640, 4168659848400, 1245471457266, 163716695587;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(8*n-18-(16*n^2-112*n+165)*x-(48*n^2-280*n+371)*x^2-(48*n^2-256*n+320)*x^3-(16*n^2-80*n+96)*x^4)*P(n-1,x) + (8-(32*n-136)*x-(128*n-456)*x^2-(192*n-632)*x^3-(128*n-400)*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[(-(8*n - 18 - (16*n^2 - 112*n + 165)*x - (48*n^2 - 280*n + 371)*x^2 - (48*n^2 - 256*n + 320)*x^3 - (16*n^2 - 80*n + 96)*x^4)*P[n - 1, x] + (8 - (32*n - 136)*x - (128*n - 456)*x^2 - (192*n - 632)*x^3 - (128*n - 400)*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] = (-(8*n - 18 - (16*n^2 - 112*n + 165)*x - (48*n^2 - 280*n + 371)*x^2 - (48*n^2 - 256*n + 320)*x^3 - (16*n^2 - 80*n + 96)*x^4)*v[i - 1] + (8 - (32*n - 136)*x - (128*n - 456)*x^2 - (192*n - 632)*x^3 - (128*n - 400)*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.