A144696 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	7	8													15
3	18	66	24												108
4	41	342	393	64											840
5	88	1436	3624	1892	160										7200
6	183	5364	25266	28788	8055	384									68040
7	374	18654	149340	312380	192270	31686	896								705600
8	757	61946	793227	2756620	3105995	1144698	118069	2048							7983360
9	1524	199464	3916260	21186552	39310620	26648856	6286476	423240	4608						97977600
10	3059	628872	18357804	147706200	421246050	469578456	205743468	32547336	1474515	10240					1297296000
11	6130	1953914	82864296	958824648	4004538300	6825516012	4932996936	1468602984	161208162	5027690	22528				18441561600
12	12273	6008862	363696765	5898980520	34814007090	86222179764	95205689586	47043691560	9872973405	771785310	16861713	49152			280215936000
13	24560	18345684	1562946576	34829121820	282621299040	979546534920	1566385861152	1185038740728	416274652080	63339490660	3598195920	55822764	106496		4533271142400
14	49135	55724732	6609527142	199152754316	2174628219265	10248281332920	22816325505876	24986590825656	13494926604033	3471898744460	391545549670	16426914492	182914927	229376	77806624896000

Technical details

Node id: n-e4ef178c3af5facc · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ce8e47375cd33c93

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:    7,      8;
  3:   18,     66,      24;
  4:   41,    342,     393,       64;
  5:   88,   1436,    3624,     1892,      160;
  6:  183,   5364,   25266,    28788,     8055,      384;
  7:  374,  18654,  149340,   312380,   192270,    31686,     896;
  8:  757,  61946,  793227,  2756620,  3105995,  1144698,  118069,   2048;
  9: 1524, 199464, 3916260, 21186552, 39310620, 26648856, 6286476, 423240, 4608;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-1)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx 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[Together[(-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
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] = (-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (3*n + 1 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); 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 2-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.