Real-rooted atlas

A075497 → D+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-96 + 32\,x + 12\,x^{2} - x^{4}\right)\,P_{n-4}(x) + \left(-32\,x + 16\,x^{2} + 4\,x^{3} - 2\,x^{4}\right)\,P_{n-4}'(x) + \left(136 - 24\,x - 10\,x^{2}\right)\,P_{n-3}(x) + \left(40\,x - 12\,x^{2} - 2\,x^{3}\right)\,P_{n-3}'(x) + \left(-68 + 4\,x + 2\,x^{2}\right)\,P_{n-2}(x) + \left(-16\,x + 2\,x^{2}\right)\,P_{n-2}'(x) + \left(14\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 8 + 2\,x\)
\(P_{3}(x) = 52 + 30\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
28210
35230385
4320320724716
519363000114014056221
61164826432151203080240656526
7699522251201824485488070003787539785

All 15 rows

Technical details
Node id
n-23d5fcfc7028d4d5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-23806764ff3107b5

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:       8,        2;
  3:      52,       30,        3;
  4:     320,      320,       72,        4;
  5:    1936,     3000,     1140,      140,       5;
  6:   11648,    26432,    15120,     3080,     240,      6;
  7:   69952,   225120,   182448,    54880,    7000,    378,     7;
  8:  419840,  1879040,  2080512,   868224,  161280,  14112,   560,   8;
  9: 2519296, 15484800, 22877760, 12747840, 3237360, 408240, 26040, 792, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(96-32*x-12*x^2+x^4)*P(n-4,x) - (32*x-16*x^2-4*x^3+2*x^4)*d/dx P(n-4,x) + (136-24*x-10*x^2)*P(n-3,x) + (40*x-12*x^2-2*x^3)*d/dx P(n-3,x) - (68-4*x-2*x^2)*P(n-2,x) - (16*x-2*x^2)*d/dx P(n-2,x) + 14*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 52 + 30*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 8 + 2*x;
P[3, x_] := 52 + 30*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(96 - 32*x - 12*x^2 + x^4)*P[n - 4, x] - (32*x - 16*x^2 - 4*x^3 + 2*x^4)*D[P[n - 4, x], x] + (136 - 24*x - 10*x^2)*P[n - 3, x] + (40*x - 12*x^2 - 2*x^3)*D[P[n - 3, x], x] - (68 - 4*x - 2*x^2)*P[n - 2, x] - (16*x - 2*x^2)*D[P[n - 2, x], x] + 14*P[n - 1, x] + 2*x*D[P[n - 1, x], 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] = 8 + 2*x; v[4] = 52 + 30*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(96 - 32*x - 12*x^2 + x^4)*v[i - 4] - (32*x - 16*x^2 - 4*x^3 + 2*x^4)*deriv(v[i - 4], x) + (136 - 24*x - 10*x^2)*v[i - 3] + (40*x - 12*x^2 - 2*x^3)*deriv(v[i - 3], x) - (68 - 4*x - 2*x^2)*v[i - 2] - (16*x - 2*x^2)*deriv(v[i - 2], x) + 14*v[i - 1] + 2*x*deriv(v[i - 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. [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.