A225467 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-21 - 24\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(-12\,x - 16\,x^{2}\right)\,P_{n-2}'(x) + \left(10 + 8\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 0\)

\(P_{2}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	4														4
3	40	32													72
4	316	672	192												1180
5	2320	9920	6912	1024											20176
6	16564	127680	163200	56320	5120										368884
7	116920	1536992	3214080	1940480	399360	24576									7232408
8	821356	17842272	57433152	54010880	18887680	2580480	114688								151690508
9	5758240	202835840	968550912	1332516864	704471040	160333824	15597568	524288							3390588576
10	40333924	2277260160	15745278720	30483425280	22865418240	7710474240	1231749120	89653248	2359296						80445952228
11	282416200	25372533152	249842741760	663159900160	678321745920	317936812032	74166435840	8768716800	495452160	10485760					2018357239784
12	1977149596	281357194272	3899876325312	13932532725760	18917199656960	11840238305280	3782864633856	645951651840	58781859840	2652897280	46137344				53363478537340
13	13840755760	3110746333760	60186288045312	285517462191104	504758255738880	410488992595968	172523249074176	39901403283456	5199274967040	375327293440	13841203200	201326592			1482088882808688
14	96887416084	34328935717440	921458798682240	5745825317872640	13037027192949760	13506042427637760	7263836020080640	2185226824974336	382328038490112	39246962360320	2301976641536	70665633792	872415232		43117790920871892
15	678218289880	378393392220512	14027855594538240	114085127965885440	328580752027207680	427240876072378368	288207114618142720	109689046394470400	24744314213498880	3386855108116480	280879536537600	13648936304640	354200584192	3758096384	1310635900036271416

Technical details

Node id: n-f1b46de67f8e4966 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-668e26fcca6da53a

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 -2)
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:        0;
   2:        4;
   3:       40,         32;
   4:      316,        672,         192;
   5:     2320,       9920,        6912,        1024;
   6:    16564,     127680,      163200,       56320,        5120;
   7:   116920,    1536992,     3214080,     1940480,      399360,      24576;
   8:   821356,   17842272,    57433152,    54010880,    18887680,    2580480,     114688;
   9:  5758240,  202835840,   968550912,  1332516864,   704471040,  160333824,   15597568,   524288;
  10: 40333924, 2277260160, 15745278720, 30483425280, 22865418240, 7710474240, 1231749120, 89653248, 2359296;

%F Formula

The row polynomials satisfy P(n,x) = -(21+24*x+16*x^2)*P(n-2,x) - (12*x+16*x^2)*d/dx P(n-2,x) + (10+8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(21 + 24*x + 16*x^2)*P[n - 2, x] - (12*x + 16*x^2)*D[P[n - 2, x], x] + (10 + 8*x)*P[n - 1, x] + 4*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] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = -(21 + 24*x + 16*x^2)*v[i - 2] - (12*x + 16*x^2)*deriv(v[i - 2], x) + (10 + 8*x)*v[i - 1] + 4*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 read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n …?
  - 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.