A131222 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-288 + 408\,n - 212\,n^{2} + 48\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(-180 + 24\,x + 222\,n - 20\,n x - 90\,n^{2} + 4\,n^{2} x + 12\,n^{3}\right)\,P_{n-3}(x) + \left(-52 + 12\,x - x^{2} + 52\,n - 6\,n x - 13\,n^{2}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 3 + 2\,x\)

\(P_{3}(x) = 14 + 18\,x + 3\,x^{2}\)

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	3	2													5
3	14	18	3												35
4	90	166	54	4											314
5	744	1740	825	120	5										3434
6	7560	20948	12825	2740	225	6									44304
7	91440	287784	212772	59220	7175	378	7								658776
8	1285200	4466712	3822588	1297156	205800	16044	588	8							11094096
9	20603520	77539680	74611716	29595888	5757045	589680	32046	864	9						208730448
10	371226240	1491571872	1580373180	711780800	162799875	20519478	1468530	58800	1215	10					4339800000
11	7428153600	31532466240	36218444328	18133328400	4737454150	704801790	62227011	3286800	100980	1650	11				98820264960
12	163459296000	727281881280	894825262584	490099320304	143186674950	24429118890	2552683518	166644984	6762690	164450	2178	12			2446007811840
13	3923502105600	18184495230720	23742962972352	14049477183456	4516656552980	865204617420	103969314449	8048856816	404273727	12998700	256399	2808	13		65394734365440
14	102017281766400	490130371249920	674066816841600	426675906183072	149037209461140	31541082849276	4267771392885	380734254472	22711679991	904833930	23618595	385476	3549	14	1878140814520320

Technical details

Node id: n-890b2f4d886ce308 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3b33a02937f0bff7

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:        3,        2;
  3:       14,       18,        3;
  4:       90,      166,       54,        4;
  5:      744,     1740,      825,      120,       5;
  6:     7560,    20948,    12825,     2740,     225,      6;
  7:    91440,   287784,   212772,    59220,    7175,    378,     7;
  8:  1285200,  4466712,  3822588,  1297156,  205800,  16044,   588,   8;
  9: 20603520, 77539680, 74611716, 29595888, 5757045, 589680, 32046, 864, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^4-48*n^3+212*n^2-408*n+288)*P(n-4,x) + (12*n^3-90*n^2+222*n-180+(4*n^2-20*n+24)*x)*P(n-3,x) - (13*n^2-52*n+52+(6*n-12)*x+x^2)*P(n-2,x) + (6*n-9+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 14 + 18*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[3, x_] := 14 + 18*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*P[n - 4, x] + (12*n^3 - 90*n^2 + 222*n - 180 + (4*n^2 - 20*n + 24)*x)*P[n - 3, x] - (13*n^2 - 52*n + 52 + (6*n - 12)*x + x^2)*P[n - 2, x] + (6*n - 9 + 2*x)*P[n - 1, 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] = 3 + 2*x; v[4] = 14 + 18*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*v[i - 4] + (12*n^3 - 90*n^2 + 222*n - 180 + (4*n^2 - 20*n + 24)*x)*v[i - 3] - (13*n^2 - 52*n + 52 + (6*n - 12)*x + x^2)*v[i - 2] + (6*n - 9 + 2*x)*v[i - 1]); 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 Exponential Riordan array [1, log((1-x)/(1-2x))].?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A079638 → θ+1

References

[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.