A269947 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-64 + 16\,x - x^{2} + 192\,n - 24\,n x - 240\,n^{2} + 12\,n^{2} x + 160\,n^{3} - 2\,n^{3} x - 60\,n^{4} + 12\,n^{5} - n^{6}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 15\,n - 9\,n^{2} + 2\,n^{3}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	1	2													3
3	8	18	3												29
4	216	502	108	4											830
5	13824	32560	7665	400	5										54454
6	1728000	4097648	1006965	60220	1125	6									6893964
7	373248000	888547968	223650912	14350140	318275	2646	7								1500117948
8	128024064000	305518449024	78045084768	5220299236	127106000	1289508	5488	8							516936298032
9	65548320768000	156681494028288	40417361074752	2776853321856	71603646045	812755296	4314282	10368	9						265496449918896
10	47784725839872000	114351905788157952	29699278464536640	2078215886399360	55670124619125	678422986038	4093326090	12488880	18225	10					193970478632404320
11	47784725839872000000	114447475239837696000	29870806323218876928	2117814924352075520	58267894477124200	745227135580950	4884819573711	17166966960	32274990	30250	11				194279840350912199520
12	63601470092869632000000	152425158995903717376000	39929714429064081735168	2858639406076904353024	80201836204492404600	1061818790830793490	7371126510787116	28431883965144	62270849520	76123850	47916	12			258896253978530400435840
13	109903340320478724096000000	263517877685107362889728000	69227184271916588814434304	4992968512939642831005696	142162072218959005590080	1931065074001002036240	13976095199942728853	57554440075525536	139589897431347	200731845600	166535083	73008	13		447785477961524709452933760
14	241457638684091756838912000000	579168583954821833716924416000	152487400661928406669646757888	11061854735290950751472093184	318571283306227488820162880	4413144454242952280327376	32958390407275344332321	142419785074435578424	371427749741635587	596107750818030	586682607511	342073212	107653	14	984498495565075635212352910080

Technical details

Node id: n-e86c758b7b6e7950 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-7db9b112e92ee086

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:              1;
  2:              1,               2;
  3:              8,              18,              3;
  4:            216,             502,            108,             4;
  5:          13824,           32560,           7665,           400,           5;
  6:        1728000,         4097648,        1006965,         60220,        1125,         6;
  7:      373248000,       888547968,      223650912,      14350140,      318275,      2646,       7;
  8:   128024064000,    305518449024,    78045084768,    5220299236,   127106000,   1289508,    5488,     8;
  9: 65548320768000, 156681494028288, 40417361074752, 2776853321856, 71603646045, 812755296, 4314282, 10368, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64+(2*n^3-12*n^2+24*n-16)*x+x^2)*P(n-2,x) + (2*n^3-9*n^2+15*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*P[n - 2, x] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*P[n - 1, x]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*v[i - 2] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*v[i - 1]); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Parent-row provers of polytool 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.