A071951 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(-12\,x - 5\,x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} - x^{3}\right)\,P_{n-2}''(x) + \left(8 + 2\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

\(P_{1}(x) = 0\)

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	1							1
3	8	2						10
4	52	40	3					95
5	320	584	120	4				1028
6	1936	7648	3276	280	5			13145
7	11648	95648	76992	12768	560	6		197622
8	69952	1171072	1683312	485696	39480	1008	7	3450527

All 15 rows

Technical details

Node id: n-a8ea30cabe0c908d · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-728f8e328f25a302

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 -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       0;
   2:       1;
   3:       8,         2;
   4:      52,        40,         3;
   5:     320,       584,       120,         4;
   6:    1936,      7648,      3276,       280,         5;
   7:   11648,     95648,     76992,     12768,       560,       6;
   8:   69952,   1171072,   1683312,    485696,     39480,    1008,      7;
   9:  419840,  14192768,  35422848,  16815296,   2265280,  103824,   1680,    8;
  10: 2519296, 171152896, 729746112, 551689344, 116160880, 8532480, 242088, 2640, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(12+4*x+x^2)*P(n-2,x) - (12*x+5*x^2)*d/dx P(n-2,x) - (2*x^2+x^3)*d^2/dx^2 P(n-2,x) + (8+2*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(12 + 4*x + x^2)*P[n - 2, x] - (12*x + 5*x^2)*D[P[n - 2, x], x] - (2*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (8 + 2*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(12 + 4*x + x^2)*v[i - 2] - (12*x + 5*x^2)*deriv(v[i - 2], x) - (2*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (8 + 2*x)*v[i - 1] + 6*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.