A166979 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(-9\,x - 9\,x^{2}\right)\,P_{n-2}'(x) + \left(11 + 2\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 11 + 2\,x\)

\(P_{2}(x) = 111 + 60\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	11	2															13
2	111	60	3														174
3	1111	1362	174	4													2651
4	11111	28100	6915	380	5												46511
5	111111	556122	235770	23280	705	6											926994
6	1111111	10788540	7435743	1175720	61530	1176	7										20573827
7	11111111	207204482	224383614	53415964	4300030	138516	1820	8									500555545
8	111111111	3959107380	6593547915	2275568820	264571335	12778416	278082	2664	9								13216965732
9	1111111111	75445262442	190558002690	92987443560	15014742435	1020298482	32705400	512280	3735	10							376170082145
10	11111111111	1435682208620	5448791968983	3694612748640	806912456460	74134107432	3283493829	74774040	882585	5060	11						11474603756771
11	111111111111	27300184186002	154719698444454	143965727658324	41736238932960	5045670856512	296633397060	9211069296	156492765	1441110	6666	12					373184633596272
12	1111111111111	518925721756260	4373101832723715	5533024854617260	2099824150489065	327595383827712	24871153411104	1011417655248	23194859688	305021860	2251821	8580	13				12879489127733437
13	11111111111111	9861810935591162	123225239898898410	210552722064470240	103508191990768565	20537535091111038	1973948432776320	102257664160080	3039823356570	53529055580	560706146	3391752	10829	14			469775911105407817
14	111111111111111	187396629998454300	3465099433572542223	7954751036250596760	5024567734155941790	1253774260400029368	150293157303980691	9720750549715920	364305387418830	8248725565080	114952575738	981380400	4952220	13440	15		18046086783394277886
15	1111111111111111	3560758192192853922	97303879085028863694	298945920919368803084	241073554566486568290	74987065602988745388	11081498704588131820	881378398476668664	40808886136774470	1155417790220980	20568855695388	232373273496	1647524060	7040040	16440	16	727877151687052290863

Technical details

Node id: n-500e32224f9dbdfe · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-dd3f1b3bb23145ea

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 0)
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:          11,             2;
   2:         111,            60,             3;
   3:        1111,          1362,           174,             4;
   4:       11111,         28100,          6915,           380,            5;
   5:      111111,        556122,        235770,         23280,          705,           6;
   6:     1111111,      10788540,       7435743,       1175720,        61530,        1176,          7;
   7:    11111111,     207204482,     224383614,      53415964,      4300030,      138516,       1820,        8;
   8:   111111111,    3959107380,    6593547915,    2275568820,    264571335,    12778416,     278082,     2664,      9;
   9:  1111111111,   75445262442,  190558002690,   92987443560,  15014742435,  1020298482,   32705400,   512280,   3735,   10;
  10: 11111111111, 1435682208620, 5448791968983, 3694612748640, 806912456460, 74134107432, 3283493829, 74774040, 882585, 5060, 11;

%F Formula

The row polynomials satisfy P(n,x) = -(10+2*x+x^2)*P(n-2,x) - (9*x+9*x^2)*d/dx P(n-2,x) + (11+2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 11 + 2*x, P(2,x) = 111 + 60*x + 3*x^2.

%t Mathematica

P[1, x_] := 11 + 2*x;
P[2, x_] := 111 + 60*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(10 + 2*x + x^2)*P[n - 2, x] - (9*x + 9*x^2)*D[P[n - 2, x], x] + (11 + 2*x)*P[n - 1, x] + 9*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] = 11 + 2*x; v[2] = 111 + 60*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = -(10 + 2*x + x^2)*v[i - 2] - (9*x + 9*x^2)*deriv(v[i - 2], x) + (11 + 2*x)*v[i - 1] + 9*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 T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m…?
  - 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.