A280580 → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(6 + 16\,x - 16\,n - 20\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 5\,x^{2} + 16\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	6	2													8
4	30	30	3												63
5	140	280	84	4											508
6	630	2100	1260	180	5										4175
7	2772	13860	13860	3960	330	6									34788
8	12012	84084	126126	60060	10010	546	7								292845
9	51480	480480	1009008	720720	200200	21840	840	8							2484576
10	218790	2625480	7351344	7351344	3063060	556920	42840	1224	9						21211011
11	923780	13856700	49884120	66512160	38798760	10581480	1356600	77520	1710	10					181992840
12	3879876	71131060	320089770	548725320	426786360	162954792	31337460	2984520	131670	2310	11				1568023149
13	16224936	356948592	1963217256	4206894120	4206894120	2141691552	576609264	82372752	6056820	212520	3036	12			13557124980
14	67603900	1757701400	11600829240	30383124200	37978905250	24858919800	8923714800	1784742960	196846650	11511500	328900	3900	13		117564232513
15	280816200	8518091400	66441112920	208814926320	319022804100	261018657900	120470149800	32125373280	4960535580	435134700	20720700	491400	4914	14	1022108819228

Technical details

Node id: n-89b33e7a8df4304e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-c613f7c62d85a96e

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:      1;
   3:      6,       2;
   4:     30,      30,       3;
   5:    140,     280,      84,       4;
   6:    630,    2100,    1260,     180,       5;
   7:   2772,   13860,   13860,    3960,     330,      6;
   8:  12012,   84084,  126126,   60060,   10010,    546,     7;
   9:  51480,  480480, 1009008,  720720,  200200,  21840,   840,    8;
  10: 218790, 2625480, 7351344, 7351344, 3063060, 556920, 42840, 1224, 9;

%F Formula

The row polynomials satisfy (2*n^2-5*n+2)*P(n,x) = (8*n^2-16*n+6+(6*n^2-20*n+16)*x)*P(n-1,x) + ((16*n-20)*x-(4*n-5)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 16*n + 6 + (6*n^2 - 20*n + 16)*x)*P[n - 1, x] + ((16*n - 20)*x - (4*n - 5)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 16*n + 6 + (6*n^2 - 20*n + 16)*x)*v[i - 1] + ((16*n - 20)*x - (4*n - 5)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 2)); 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) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

References

[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.