Real-rooted atlas

A130534 → D → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-8 + 12\,x - 6\,x^{2} + x^{3} + 12\,n - 12\,n x + 3\,n x^{2} - 6\,n^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n-3}(x) + \left(-7 + 9\,x - 3\,x^{2} + 9\,n - 6\,n x - 3\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 3\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2527
32318344
4120130424296
57249604358052204
65012765843051120135618236
73933266878438271428024502107166984

All 15 rows

Technical details
Node id
n-939617ce78b89743 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b4083b2ff491c73b

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       1;
  2:       5,       2;
  3:      23,      18,       3;
  4:     120,     130,      42,       4;
  5:     724,     960,     435,      80,      5;
  6:    5012,    7658,    4305,    1120,    135,     6;
  7:   39332,   66878,   43827,   14280,   2450,   210,    7;
  8:  345832,  639952,  471240,  180516,  39060,  4788,  308,   8;
  9: 3371976, 6687480, 5402940, 2342760, 599865, 93240, 8610, 432, 9;
%F Formula
The row polynomials satisfy P(n,x) = (n^3-6*n^2+12*n-8+(3*n^2-12*n+12)*x+(3*n-6)*x^2+x^3)*P(n-3,x) - (3*n^2-9*n+7+(6*n-9)*x+3*x^2)*P(n-2,x) + (3*n-3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[(n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*P[n - 2, x] + (3*n - 3 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 5 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*v[i - 3] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*v[i - 2] + (3*n - 3 + 3*x)*v[i - 1]); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.