A111577 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - 3\,x^{2}\right)\,P_{n-2}'(x) + \left(5 + 2\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

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

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	5	2													7
4	21	24	3												48
5	85	210	66	4											365
6	341	1640	975	140	5										3101
7	1365	12162	12210	3120	255	6									29118
8	5461	87864	140343	56840	7980	420	7								298915
9	21845	625970	1535226	926044	198730	17556	644	8							3326023
10	87381	4425480	16291215	14085540	4337235	572040	34650	936	9						39834486
11	349525	31153122	169550370	204833640	87002685	16073442	1429680	63000	1305	10					510456779
12	1398101	218770904	1742233383	2888904480	1648085010	409798620	50205309	3208920	107415	1760	11				6962713913
13	5592405	1534192530	17750490186	39878736084	29980490760	9763875792	1582615188	137600496	6617655	173910	2310	12			100640387328
14	22369621	10750532520	179806190655	542090889340	529536272265	221490228960	46208722560	5248715472	340094898	12744160	269841	2964	13		1535507033269
15	89478485	75298466882	1814187705330	7286923148960	9150193967915	4843757876958	1274997163440	184027855440	15427462050	772952180	23193170	404040	3731	14	24645699678595

Technical details

Node id: n-875f8968dcda360b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-814590f13101ba28

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:     5,       2;
   4:    21,      24,        3;
   5:    85,     210,       66,        4;
   6:   341,    1640,      975,      140,       5;
   7:  1365,   12162,    12210,     3120,     255,      6;
   8:  5461,   87864,   140343,    56840,    7980,    420,     7;
   9: 21845,  625970,  1535226,   926044,  198730,  17556,   644,   8;
  10: 87381, 4425480, 16291215, 14085540, 4337235, 572040, 34650, 936, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(4+2*x+x^2)*P(n-2,x) - (3*x+3*x^2)*d/dx P(n-2,x) + (5+2*x)*P(n-1,x) + 3*x*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[-(4 + 2*x + x^2)*P[n - 2, x] - (3*x + 3*x^2)*D[P[n - 2, x], x] + (5 + 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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(4 + 2*x + x^2)*v[i - 2] - (3*x + 3*x^2)*deriv(v[i - 2], x) + (5 + 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 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 Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.?
  - 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.