A269939 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1	1														1
2	1														1
3	3	1													4
4	15	10	1												26
5	105	105	25	1											236
6	945	1260	490	56	1										2752
7	10395	17325	9450	1918	119	1									39208
8	135135	270270	190575	56980	6825	246	1								660032
9	2027025	4729725	4099095	1636635	302995	22935	501	1							12818912
10	34459425	91891800	94594500	47507460	12122110	1487200	74316	1012	1						282137824
11	654729075	1964187225	2343240900	1422280860	466876410	81431350	6914908	235092	2035	1					6939897856
12	13749310575	45831035250	62199262125	44346982680	17892864990	4104160060	510880370	30950920	731731	4082	1				188666182784
13	316234143225	1159525191825	1764494857125	1446733012725	694740296250	199124936010	33309926650	3049616570	134779645	2252341	8177	1			5617349020544
14	7905853580625	31623414322500	53338158823950	49473074851200	27610143335775	9540421090200	2026763158420	254752658160	17539336815	575156036	6879678	16368	1		181790703209728
15	213458046676875	924984868933125	1712934942468750	1774073543492250	1129764045234825	458380554006375	118885395048420	19282395272140	1861763348445	98049492723	2417578670	20900922	32751	1	6353726042486272

Technical details

Node id: n-4f3d85d3aa6073ea · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-014ae34d4ff839c2

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:        1;
   3:        3,        1;
   4:       15,       10,        1;
   5:      105,      105,       25,        1;
   6:      945,     1260,      490,       56,        1;
   7:    10395,    17325,     9450,     1918,      119,       1;
   8:   135135,   270270,   190575,    56980,     6825,     246,     1;
   9:  2027025,  4729725,  4099095,  1636635,   302995,   22935,   501,    1;
  10: 34459425, 91891800, 94594500, 47507460, 12122110, 1487200, 74316, 1012, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 3 + (n - 2)*x)*P[n - 1, x] - (x + x^2)*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; for(n = 2, N, my(i = n); v[i] = (2*n - 3 + (n - 2)*x)*v[i - 1] - (x + x^2)*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m),…?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.