A151775 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 24\,x - 24\,x^{2}\right)\,P_{n-2}(x) + \left(-6 - 32\,x - 62\,x^{2} - 36\,x^{3}\right)\,P_{n-2}'(x) + \left(-4\,x - 16\,x^{2} - 20\,x^{3} - 8\,x^{4}\right)\,P_{n-2}''(x) + \left(14 + 24\,x\right)\,P_{n-1}(x) + \left(6 + 28\,x + 22\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x + 8\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	2							2
2	28	48						76
3	662	2640	2160					5462
4	24568	167328	302400	161280				655576
5	1326122	13498080	40007520	45964800	18144000			118940522
6	98329108	1385398608	5938574400	10958492160	9180864000	2874009600		30435667876
7	9596075582	177932729520	1027378432080	2693417126400	3577942368000	2353813862400	610248038400	10450328632382

All 15 rows

Technical details

Node id: n-789f42810b77fd52 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3764fa9cd037f7ff

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:              28,               48;
  3:             662,             2640,              2160;
  4:           24568,           167328,            302400,             161280;
  5:         1326122,         13498080,          40007520,           45964800,           18144000;
  6:        98329108,       1385398608,        5938574400,        10958492160,         9180864000,         2874009600;
  7:      9596075582,     177932729520,     1027378432080,      2693417126400,      3577942368000,      2353813862400,       610248038400;
  8:   1192744081648,   28073116365888,   208457658864000,    724868256752640,   1352654214912000,   1398688504012800,    756707567616000,    167382319104000;
  9: 183983154281042, 5348776806240960, 49426658023515840, 218061839231308800, 534339995821632000, 770096309946163200, 650712365330227200, 298777439600640000, 57621363351552000;

%F Formula

The row polynomials satisfy P(n,x) = -(9+24*x+24*x^2)*P(n-2,x) - (6+32*x+62*x^2+36*x^3)*d/dx P(n-2,x) - (4*x+16*x^2+20*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (14+24*x)*P(n-1,x) + (6+28*x+22*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(9 + 24*x + 24*x^2)*P[n - 2, x] - (6 + 32*x + 62*x^2 + 36*x^3)*D[P[n - 2, x], x] - (4*x + 16*x^2 + 20*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (14 + 24*x)*P[n - 1, x] + (6 + 28*x + 22*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}]];
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; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 24*x + 24*x^2)*v[i - 2] - (6 + 32*x + 62*x^2 + 36*x^3)*deriv(v[i - 2], x) - (4*x + 16*x^2 + 20*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (14 + 24*x)*v[i - 1] + (6 + 28*x + 22*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*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 - 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.?
  - 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.