A166973 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 + 3\,x + 4\,x^{2}\right)\,P_{n-2}(x) + \left(-5\,x + 20\,x^{2}\right)\,P_{n-2}'(x) + \left(7 - 3\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	8	9	1													18
4	44	79	21	1												145
5	260	741	343	38	1											1383
6	1556	7375	5265	1005	60	1										15262
7	9332	76461	79975	23230	2345	87	1									191431
8	55988	813079	1221801	505925	76440	4718	119	1								2678071
9	335924	8775909	18847711	10721886	2268525	206514	8554	156	1							41165180
10	2015540	95527231	293459313	224503693	63857556	8024835	484386	14358	198	1						687887111
11	12093236	1044752925	4603852855	4679098910	1742530945	289995237	23884875	1021680	22710	245	1					12397253619
12	72559412	11456002471	72641079225	97414899285	46668312240	10001676784	1078304997	62346075	1983960	34265	297	1				239397199012
13	435356468	125798348949	1150946383951	2029088153950	1235692570805	334405332096	45872032228	3439551852	146802975	3605855	49753	354	1			4925828189237
14	2612138804	1382475769039	18290214507201	42310777766621	32490761803500	10946970447413	1872076173968	176927113792	9718530822	318117800	6208059	69979	416	1		107482858647415
15	15672832820	15199397043021	291266180623783	883349504016798	850626567788361	353143039604355	74161033423477	8653569923856	594580714156	24889726862	643803160	10216206	95823	483	1	2477035089813162

Technical details

Node id: n-95486bda0555a888 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c93feb6ff1f2d417

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:       1;
   2:       2,        1;
   3:       8,        9,         1;
   4:      44,       79,        21,         1;
   5:     260,      741,       343,        38,        1;
   6:    1556,     7375,      5265,      1005,       60,       1;
   7:    9332,    76461,     79975,     23230,     2345,      87,      1;
   8:   55988,   813079,   1221801,    505925,    76440,    4718,    119,     1;
   9:  335924,  8775909,  18847711,  10721886,  2268525,  206514,   8554,   156,   1;
  10: 2015540, 95527231, 293459313, 224503693, 63857556, 8024835, 484386, 14358, 198, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(6-3*x-4*x^2)*P(n-2,x) - (5*x-20*x^2)*d/dx P(n-2,x) + (7-3*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[-(6 - 3*x - 4*x^2)*P[n - 2, x] - (5*x - 20*x^2)*D[P[n - 2, x], x] + (7 - 3*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = -(6 - 3*x - 4*x^2)*v[i - 2] - (5*x - 20*x^2)*deriv(v[i - 2], x) + (7 - 3*x)*v[i - 1] + 5*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, 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m…?
  - 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.