Real-rooted atlas

A074246 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
212618
3706012142
4450510180201160
53248441021004203010208
626264406142352064408404297720
723624840370426938890720163801512561018008

All 15 rows

Technical details
Node id
n-fd9c1c0b6460954a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-1db1b1b28f935ec6

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:        2;
  2:       12,        6;
  3:       70,       60,       12;
  4:      450,      510,      180,       20;
  5:     3248,     4410,     2100,      420,      30;
  6:    26264,    40614,    23520,     6440,     840,     42;
  7:   236248,   403704,   269388,    90720,   16380,   1512,    56;
  8:  2345400,  4342080,  3231900,  1265460,  283500,  36540,  2520,   72;
  9: 25507152, 50457000, 41003160, 18041100, 4733190, 762300, 73920, 3960, 90;
%F Formula
The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n-1+(3*n^2-6*n+3)*x+(3*n-3)*x^2+x^3)*P(n-3,x) - (3*n^2-3*n+1+(6*n-3)*x+3*x^2)*P(n-2,x) + (3*n+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 12 + 6*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 12 + 6*x;
P[n_, x_] := P[n, x] = Expand[(n^3 - 3*n^2 + 3*n - 1 + (3*n^2 - 6*n + 3)*x + (3*n - 3)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 3*n + 1 + (6*n - 3)*x + 3*x^2)*P[n - 2, x] + (3*n + 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] = 2; v[3] = 12 + 6*x; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 3*n^2 + 3*n - 1 + (3*n^2 - 6*n + 3)*x + (3*n - 3)*x^2 + x^3)*v[i - 3] - (3*n^2 - 3*n + 1 + (6*n - 3)*x + 3*x^2)*v[i - 2] + (3*n + 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.