A285061 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-5 - 8\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(-4\,x - 16\,x^{2}\right)\,P_{n-2}'(x) + \left(6 + 8\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 0\)

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	4							4
2	24	32						56
3	124	480	192					796
4	624	5312	5376	1024				12336
5	3124	52800	101760	46080	5120			208884
6	15624	500192	1639680	1326080	337920	24576		3844072
7	78124	4626720	24316992	31288320	13726720	2236416	114688	76387980

All 15 rows

Technical details

Node id: n-b081b98b73e99863 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-7af788cb0422c847

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:       4;
  2:      24,        32;
  3:     124,       480,        192;
  4:     624,      5312,       5376,        1024;
  5:    3124,     52800,     101760,       46080,        5120;
  6:   15624,    500192,    1639680,     1326080,      337920,      24576;
  7:   78124,   4626720,   24316992,    31288320,    13726720,    2236416,    114688;
  8:  390624,  42265472,  343881216,   661592064,   444702720,  121798656,  13762560,   524288;
  9: 1953124, 383514240, 4724048640, 13081098240, 12646717440, 5179539456, 967507968, 80216064, 2359296;

%F Formula

The row polynomials satisfy P(n,x) = -(5+8*x+16*x^2)*P(n-2,x) - (4*x+16*x^2)*d/dx P(n-2,x) + (6+8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(5 + 8*x + 16*x^2)*P[n - 2, x] - (4*x + 16*x^2)*D[P[n - 2, x], x] + (6 + 8*x)*P[n - 1, x] + 4*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = -(5 + 8*x + 16*x^2)*v[i - 2] - (4*x + 16*x^2)*deriv(v[i - 2], x) + (6 + 8*x)*v[i - 1] + 4*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.