A144699 → D

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-8 - 16\,x - 24\,x^{2} - 2\,n - 12\,n x - 26\,n x^{2} - 2\,n^{2} x - 9\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x - 8\,x^{2} + 12\,x^{3} - n x - 6\,n x^{2} + 7\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(10 + 18\,x + 3\,n + 12\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 5\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	5														5
2	16	50													66
3	39	342	375												756
4	86	1572	4578	2500											8736
5	181	6092	34818	49124	15625										105840
6	372	21534	212376	570588	462180	93750									1360800
7	755	72042	1140213	5118284	7765725	3983946	546875								18627840
8	1522	232736	5641482	39273976	97777190	93119712	32262622	3125000							271434240
9	3057	734736	26406072	271589640	1028495370	1590499008	1018578624	249354408	17578125						4203239040
10	6128	2283690	118849536	1745237256	9565842720	22241243052	22993286400	10392818664	1858923504	97656250					69016147200
11	12271	7022654	519930099	10627778856	81392068590	270478814004	417479343246	303759700776	100432682019	13467042110	537109375				1198701504000
12	24558	21436092	2227196130	62151015996	647769094380	2967727003416	6495420909348	7027910551800	3738164127174	929280545580	95327790426	2929687500			21968929382400
13	49133	65094132	9391096590	352329407740	4896568576215	30101963982984	90049197151956	137484576787320	108539235608715	43450211576260	8299393686078	662049656652	15869140625		423860851814400
14	98284	196952918	39126645528	1949468970500	35545176092740	287100217669842	1141821072977424	2369408240996184	2632093844944500	1563344683426010	481954586104024	71983833684708	4525490736988	85449218750	8589851388518400

Technical details

Node id: n-5018307f6d50b2b7 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-136430c2d407c49e

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 -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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    0;
  1:    5;
  2:   16,     50;
  3:   39,    342,      375;
  4:   86,   1572,     4578,      2500;
  5:  181,   6092,    34818,     49124,      15625;
  6:  372,  21534,   212376,    570588,     462180,      93750;
  7:  755,  72042,  1140213,   5118284,    7765725,    3983946,     546875;
  8: 1522, 232736,  5641482,  39273976,   97777190,   93119712,   32262622,   3125000;
  9: 3057, 734736, 26406072, 271589640, 1028495370, 1590499008, 1018578624, 249354408, 17578125;

%F Formula

The row polynomials satisfy (n+3)*P(n,x) = -(2*n+8+(2*n^2+12*n+16)*x+(n^3+9*n^2+26*n+24)*x^2)*P(n-2,x) - ((n+4)*x+(n^2+6*n+8)*x^2-(n^2+7*n+12)*x^3)*d/dx P(n-2,x) + (3*n+10+(2*n^2+12*n+18)*x)*P(n-1,x) + ((n+3)*x-(n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 8 + (2*n^2 + 12*n + 16)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*P[n - 2, x] - ((n + 4)*x + (n^2 + 6*n + 8)*x^2 - (n^2 + 7*n + 12)*x^3)*D[P[n - 2, x], x] + (3*n + 10 + (2*n^2 + 12*n + 18)*x)*P[n - 1, x] + ((n + 3)*x - (n + 3)*x^2)*D[P[n - 1, x], x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 8 + (2*n^2 + 12*n + 16)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*v[i - 2] - ((n + 4)*x + (n^2 + 6*n + 8)*x^2 - (n^2 + 7*n + 12)*x^3)*deriv(v[i - 2], x) + (3*n + 10 + (2*n^2 + 12*n + 18)*x)*v[i - 1] + ((n + 3)*x - (n + 3)*x^2)*deriv(v[i - 1], x))/(n + 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of 5-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[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.