A144697 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3														3
2	10	18													28
3	25	134	81												240
4	56	652	1128	324											2160
5	119	2628	9402	7636	1215										21000
6	246	9550	61260	100860	45470	4374									221760
7	501	32586	345315	994380	903195	248874	15309								2540160
8	1012	106776	1772292	8194840	12877020	7161288	1283884	52488							31449600
9	2035	340568	8530644	59878648	148966970	142843560	52045252	6341576	177147						419126400
10	4082	1066474	39231096	401760968	1492588300	2251227468	1416266936	354475624	30308562	590490					5987520000
11	8177	3297390	174521205	2531886120	13475340690	30087439956	29715159810	12901825320	2297011005	141200110	1948617				91329638400
12	16368	10104772	757437840	15219074940	112500620640	356145759144	518537384736	353837706360	110057430960	14314373460	644663888	6377292			1482030950400
13	32751	30772620	3226794414	88214380140	884266004265	3843027761688	7887829548900	7925452589112	3887046247185	891174314460	86465020686	2895983580	20726199		25499650176000
14	65518	93300390	13553402676	497003003876	6628811727690	38573505588114	107937960277560	152461034376408	110199535773858	40035499381610	6919330723668	509275172100	12839428966	66961566	463788509184000

Technical details

Node id: n-cd9f9b6ab54e9772 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6d6ad7c2e7725e78

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:    3;
  2:   10,     18;
  3:   25,    134,      81;
  4:   56,    652,    1128,      324;
  5:  119,   2628,    9402,     7636,      1215;
  6:  246,   9550,   61260,   100860,     45470,      4374;
  7:  501,  32586,  345315,   994380,    903195,    248874,    15309;
  8: 1012, 106776, 1772292,  8194840,  12877020,   7161288,  1283884,   52488;
  9: 2035, 340568, 8530644, 59878648, 148966970, 142843560, 52045252, 6341576, 177147;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (2*n^2 + 4*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*P[n - 2, x] - ((n + 2)*x + (n^2 + 2*n)*x^2 - (n^2 + 3*n + 2)*x^3)*D[P[n - 2, x], x] + (3*n + 4 + (2*n^2 + 4*n + 2)*x)*P[n - 1, x] + ((n + 1)*x - (n + 1)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (2*n^2 + 4*n)*x + (n^3 + 3*n^2 + 2*n)*x^2)*v[i - 2] - ((n + 2)*x + (n^2 + 2*n)*x^2 - (n^2 + 3*n + 2)*x^3)*deriv(v[i - 2], x) + (3*n + 4 + (2*n^2 + 4*n + 2)*x)*v[i - 1] + ((n + 1)*x - (n + 1)*x^2)*deriv(v[i - 1], x))/(n + 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of 3-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.