A160562 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(-16\,x - 12\,x^{2}\right)\,P_{n-2}'(x) + \left(-4\,x^{2} - 4\,x^{3}\right)\,P_{n-2}''(x) + \left(10 + 2\,x\right)\,P_{n-1}(x) + \left(16\,x\right)\,P_{n-1}'(x) + \left(4\,x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	10	2													12
3	91	70	3												164
4	820	1932	252	4											3008
5	7381	49940	15246	660	5										73232
6	66430	1263262	821964	73788	1430	6									2226880
7	597871	31714410	42171129	7072780	265265	2730	7								81824192
8	5380840	794055992	2113956936	629123352	40938040	779688	4760	8							3584239616
9	48427561	19862161480	104774973852	53777600760	5739907030	180892920	1980636	7752	9						184385952000
10	435848050	496650892122	5163766960968	4495685626696	761750751580	37458791916	656684840	4503912	11970	10					10956410072064
11	3922632451	12417143999150	253769557425615	371035558377000	97791447974550	7244636735700	191456012902	2052127528	9388071	17710	11				742455784690688
12	35303692060	310436445243652	12453334029853860	30392239638437820	12296559652891800	1341693345902760	51529679094600	811871727480	5697737100	18241300	25300	12			56846645682847744
13	317733228541	7760981738475420	610679022130704618	2478375856086601900	1525874017547955075	241502047041036600	13159486699838220	293521991064120	2970238786515	14375232300	33449130	35100	13		4877648235616407552
14	2859599056870	194025178928342582	29934913557012239412	201562683039189026724	187728725943410816450	42644894770992731490	3242636895678142200	99867268788774360	1402468441879050	9639742762650	33507345300	58427460	47502	14	465409161655349592064

Technical details

Node id: n-e5e4323c11e2a9a6 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-abe2f5030b54dabd

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:       10,           2;
  3:       91,          70,            3;
  4:      820,        1932,          252,           4;
  5:     7381,       49940,        15246,         660,          5;
  6:    66430,     1263262,       821964,       73788,       1430,         6;
  7:   597871,    31714410,     42171129,     7072780,     265265,      2730,       7;
  8:  5380840,   794055992,   2113956936,   629123352,   40938040,    779688,    4760,    8;
  9: 48427561, 19862161480, 104774973852, 53777600760, 5739907030, 180892920, 1980636, 7752, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(9+2*x+x^2)*P(n-2,x) - (16*x+12*x^2)*d/dx P(n-2,x) - (4*x^2+4*x^3)*d^2/dx^2 P(n-2,x) + (10+2*x)*P(n-1,x) + 16*x*d/dx P(n-1,x) + 4*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(9 + 2*x + x^2)*P[n - 2, x] - (16*x + 12*x^2)*D[P[n - 2, x], x] - (4*x^2 + 4*x^3)*D[P[n - 2, x], {x, 2}] + (10 + 2*x)*P[n - 1, x] + 16*x*D[P[n - 1, x], x] + 4*x^2*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 2*x + x^2)*v[i - 2] - (16*x + 12*x^2)*deriv(v[i - 2], x) - (4*x^2 + 4*x^3)*deriv(deriv(v[i - 2], x), x) + (10 + 2*x)*v[i - 1] + 16*x*deriv(v[i - 1], x) + 4*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 - 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 of scaled central factorial numbers, T(n,k) = A008958(n,n-k).?
  - 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.