A285061 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	4														4
2	24	32													56
3	124	480	192												796
4	624	5312	5376	1024											12336
5	3124	52800	101760	46080	5120										208884
6	15624	500192	1639680	1326080	337920	24576									3844072
7	78124	4626720	24316992	31288320	13726720	2236416	114688								76387980
8	390624	42265472	343881216	661592064	444702720	121798656	13762560	524288							1628917600
9	1953124	383514240	4724048640	13081098240	12646717440	5179539456	967507968	80216064	2359296						37066954468
10	9765624	3467253152	63713717760	247573596160	330986557440	190192730112	52228915200	7070023680	448266240	10485760					895701311128
11	48828124	31283403360	849081849792	4548557629440	8188585687040	6343553728512	2402204614656	472071536640	48400957440	2422210560	46137344				22886256582908
12	244140624	281941255232	11225764467456	81853916232704	194703087575040	197894054510592	99267184558080	26559867518976	3915157340160	314425999360	12733906944	201326592			616028578831760
13	1220703124	2539424422080	147626585608320	1451387319782400	4498034420239360	5881926183124992	3802253939900416	1330268471820288	264380225421312	30292165263360	1956500209664	65431142400	872415232		17410732760052948
14	6103515624	22864585423712	1934382159440640	25460926226211840	101715659423938560	168638719795273728	137714353111695360	61280591866757120	15768276463779840	2417001999892480	221328036593664	11743581437952	329772957696	3758096384	515186186885014600

Technical details

Node id: n-b081b98b73e99863 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-7af788cb0422c847

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:       4;
  2:      24,        32;
  3:     124,       480,        192;
  4:     624,      5312,       5376,        1024;
  5:    3124,     52800,     101760,       46080,        5120;
  6:   15624,    500192,    1639680,     1326080,      337920,      24576;
  7:   78124,   4626720,   24316992,    31288320,    13726720,    2236416,    114688;
  8:  390624,  42265472,  343881216,   661592064,   444702720,  121798656,  13762560,   524288;
  9: 1953124, 383514240, 4724048640, 13081098240, 12646717440, 5179539456, 967507968, 80216064, 2359296;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(5 + 8*x + 16*x^2)*P[n - 2, x] - (4*x + 16*x^2)*D[P[n - 2, x], x] + (6 + 8*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], 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] = 4; for(n = 2, N, my(i = n + 1); v[i] = -(5 + 8*x + 16*x^2)*v[i - 2] - (4*x + 16*x^2)*deriv(v[i - 2], x) + (6 + 8*x)*v[i - 1] + 4*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 - 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 Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).?
  - 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.