A269944 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 8\,x - x^{2} + 32\,n + 8\,n x - 24\,n^{2} - 2\,n^{2} x + 8\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(5 + 2\,x - 6\,n + 2\,n^{2}\right)\,P_{n-1}(x)\)

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

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

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	4	10	3												17
4	36	98	42	4											180
5	576	1640	819	120	5										3160
6	14400	42152	22935	4092	275	6									83860
7	518400	1546272	888888	177892	15015	546	7								3147020
8	25401600	76804128	45874920	9901892	958100	44772	980	8							158986400
9	1625702400	4966267392	3051201072	694887648	73695765	4015128	114954	1632	9						10415886000
10	131681894400	405519063552	254596687920	60354167584	6837966525	413660286	13995590	263568	2565	10					859417702000
11	13168189440000	40815270144000	26067947387328	6374879008960	759239361980	49571588430	1882162667	42351760	553014	3850	11				87237022002000
12	1593350922240000	4964984066304000	3215444539082688	806117623267264	99836561560780	6909249434406	285575202542	7275606008	114560424	1080310	5566	12			10686935928344000
13	229442532802560000	718144407392256000	470471489727363072	120368197135929600	15384111893836400	1114735792427400	49183620172855	1374058925200	24681757815	282854000	1989845	7800	13		1554976057390080000
14	38775788043632640000	121825289914896384000	80586898375012743168	20969520635608586496	2750375156478263600	206851283192834280	9612556900377795	288425809984920	5717033361585	75226501350	647423205	3488940	10647	14	265124630184442600000

Technical details

Node id: n-6588fb5e01ae9659 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-86f4018c41b49fc2

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:          4,         10,          3;
  4:         36,         98,         42,         4;
  5:        576,       1640,        819,       120,        5;
  6:      14400,      42152,      22935,      4092,      275,       6;
  7:     518400,    1546272,     888888,    177892,    15015,     546,      7;
  8:   25401600,   76804128,   45874920,   9901892,   958100,   44772,    980,    8;
  9: 1625702400, 4966267392, 3051201072, 694887648, 73695765, 4015128, 114954, 1632, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(2*n^2-8*n+8)*x+x^2)*P(n-2,x) + (2*n^2-6*n+5+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^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (2*n^2 - 8*n + 8)*x + x^2)*P[n - 2, x] + (2*n^2 - 6*n + 5 + 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^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (2*n^2 - 8*n + 8)*x + x^2)*v[i - 2] + (2*n^2 - 6*n + 5 + 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 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
  - Parent-row provers of polytool polytool ? step › direct check
  - 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

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.