Real-rooted atlas

A397048 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5\,x + 11\,x^{2} + 8\,x^{3} + 2\,x^{4} + 2\,n - 9\,n x^{2} - 10\,n x^{3} - 3\,n x^{4} - n^{2} - 2\,n^{2} x + 2\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}(x) + \left(6\,x + 22\,x^{2} + 30\,x^{3} + 18\,x^{4} + 4\,x^{5} + n + 2\,n x - 2\,n x^{2} - 8\,n x^{3} - 7\,n x^{4} - 2\,n x^{5}\right)\,P_{n-1}'(x) + \left(x + 5\,x^{2} + 10\,x^{3} + 10\,x^{4} + 5\,x^{5} + x^{6}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
24131027
32711816777389
42561393279424557997697
531252027652044661514168410427193707
6466563509291091556179860416568918095361639675918139
7823543703841025642161516451616212905944657978177643473017562212718221

All 15 rows

Technical details
Node id
n-4322d1185c7ae1bd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-be89ca340c8a7532

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:         1,          2;
  2:         4,         13,          10;
  3:        27,        118,         167,          77;
  4:       256,       1393,        2794,        2455,         799;
  5:      3125,      20276,       52044,       66151,       41684,        10427;
  6:     46656,     350929,     1091556,     1798604,     1656891,       809536,      163967;
  7:    823543,    7038410,    25642161,    51645161,    62129059,     44657978,    17764347,     3017562;
  8:  16777216,  160487233,   668990470,  1587688661,  2346948311,   2213252537,  1300570552,   435483441,    63625324;
  9: 387420489, 4099363912, 19220457782, 52420755479, 91665170435, 106592295601, 82437769924, 40894475467, 11808426803, 1512354975;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n^2-2*n+(2*n^2-5)*x+(9*n-11)*x^2-(2*n^2-10*n+8)*x^3-(n^2-3*n+2)*x^4)*P(n-1,x) + (n+(2*n+6)*x-(2*n-22)*x^2-(8*n-30)*x^3-(7*n-18)*x^4-(2*n-4)*x^5)*d/dx P(n-1,x) + (x+5*x^2+10*x^3+10*x^4+5*x^5+x^6)*d^2/dx^2 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[(-(n^2 - 2*n + (2*n^2 - 5)*x + (9*n - 11)*x^2 - (2*n^2 - 10*n + 8)*x^3 - (n^2 - 3*n + 2)*x^4)*P[n - 1, x] + (n + (2*n + 6)*x - (2*n - 22)*x^2 - (8*n - 30)*x^3 - (7*n - 18)*x^4 - (2*n - 4)*x^5)*D[P[n - 1, x], x] + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + x^6)*D[P[n - 1, x], {x, 2}])/(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] = (-(n^2 - 2*n + (2*n^2 - 5)*x + (9*n - 11)*x^2 - (2*n^2 - 10*n + 8)*x^3 - (n^2 - 3*n + 2)*x^4)*v[i - 1] + (n + (2*n + 6)*x - (2*n - 22)*x^2 - (8*n - 30)*x^3 - (7*n - 18)*x^4 - (2*n - 4)*x^5)*deriv(v[i - 1], x) + (x + 5*x^2 + 10*x^3 + 10*x^4 + 5*x^5 + 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 - 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.