A225468 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

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	2														4
3	4	14	3													21
4	8	78	45	4												135
5	16	406	477	104	5											1008
6	32	2062	4425	1780	200	6										8505
7	64	10374	38493	25480	5025	342	7									79785
8	128	51998	323505	331604	102200	11844	539	8								821826
9	256	260246	2666037	4078984	1845305	323988	24598	800	9							9200223
10	512	1301742	21718665	48423540	30933000	7722162	869946	46512	1134	10						111017223
11	1024	6509734	175701933	561617160	493591425	168396354	26408109	2064000	81810	1550	11					1434373110
12	2048	32550718	1415380065	6412058004	7612301400	3455047728	724624593	77652696	4449060	135850	2057	12				19734204231
13	4096	162757686	11371866597	72419811464	114587292105	67870573056	18523380876	2614154400	203034843	8883050	215259	2664	13			287761976109
14	8192	813796622	91219069305	811780414900	1694746853800	1291304492478	449649952752	81295129344	8219829618	483202720	16659643	328068	3380	14		4429529740836
15	16384	4068999494	730973249373	9051209989640	24741181471825	23985872596686	10499520962931	2383673635200	305172590580	23146022900	1064631568	29656536	483847	4214	15	71725914311193

Technical details

Node id: n-e9807d8de7393cf6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-bc9ef0e3331cfb9e

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,       2;
   3:   4,      14,        3;
   4:   8,      78,       45,        4;
   5:  16,     406,      477,      104,        5;
   6:  32,    2062,     4425,     1780,      200,       6;
   7:  64,   10374,    38493,    25480,     5025,     342,      7;
   8: 128,   51998,   323505,   331604,   102200,   11844,    539,     8;
   9: 256,  260246,  2666037,  4078984,  1845305,  323988,  24598,   800,    9;
  10: 512, 1301742, 21718665, 48423540, 30933000, 7722162, 869946, 46512, 1134, 10;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x - x^2)*P[n - 2, x] + (3*x - 3*x^2)*D[P[n - 2, x], x] + (1 + 2*x)*P[n - 1, x] + 3*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 + 2*x; for(n = 3, N, my(i = n); v[i] = (2 + 2*x - x^2)*v[i - 2] + (3*x - 3*x^2)*deriv(v[i - 2], x) + (1 + 2*x)*v[i - 1] + 3*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - 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.