A075499 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-32 - 8\,x - x^{2}\right)\,P_{n-2}(x) + \left(-16\,x - 4\,x^{2}\right)\,P_{n-2}'(x) + \left(12 + 2\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 12 + 2\,x\)

\(P_{1}(x) = 112 + 48\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	12	2															14
1	112	48	3														163
2	960	800	120	4													1884
3	7936	11520	3120	240	5												22821
4	64512	154112	67200	8960	420	6											295210
5	520192	1978368	1306368	268800	21280	672	7										4095687
6	4177920	24780800	23869440	7117824	846720	44352	1008	8									60838072
7	33488896	305725440	419082240	174182400	29218560	2257920	84000	1440	9								964040905
8	268173312	3735683072	7163904000	4042424320	918973440	98284032	5322240	147840	1980	10							16232914246
9	2146435072	45364543488	120225988608	90400358400	27108392960	3854721024	284976384	11404800	245520	2640	11						289397068907
10	17175674880	548667392000	1991662632960	1968308486144	763601879040	140462260224	13616418816	736260096	22651200	388960	3432	12					5444254047764
11	137422176256	6618360053760	32689603215360	42021719900160	20786830704640	4849265541120	599598039040	42066984960	1734340608	42282240	592592	4368	13				107746643835117
12	1099444518912	79695164997632	532961191526400	884020535623680	551411086786560	160723631996928	24844613713920	2199666360320	116698982400	3787463680	74954880	873600	5460	14			2237075897804386
13	8795824586752	958540869009408	8646921811918848	18391025634508800	14338891752407040	5161955000057856	982538542841856	107581833216000	7142583951360	296313937920	7764044288	127196160	1252160	6720	15		48603698058935183
14	70367670435840	11520082077286400	139788560294215680	379349741772734464	367122184100904960	161741410104508416	37463514204340224	4995847187595264	406732920422400	20974017658880	698619457536	15084066816	207916800	1751680	8160	16	1102480128263303536

Technical details

Node id: n-ab32dd2a3541f242 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-cea55ac695ee84a8

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:         12,           2;
  1:        112,          48,            3;
  2:        960,         800,          120,           4;
  3:       7936,       11520,         3120,         240,           5;
  4:      64512,      154112,        67200,        8960,         420,          6;
  5:     520192,     1978368,      1306368,      268800,       21280,        672,         7;
  6:    4177920,    24780800,     23869440,     7117824,      846720,      44352,      1008,        8;
  7:   33488896,   305725440,    419082240,   174182400,    29218560,    2257920,     84000,     1440,      9;
  8:  268173312,  3735683072,   7163904000,  4042424320,   918973440,   98284032,   5322240,   147840,   1980,   10;
  9: 2146435072, 45364543488, 120225988608, 90400358400, 27108392960, 3854721024, 284976384, 11404800, 245520, 2640, 11;

%F Formula

The row polynomials satisfy P(n,x) = -(32+8*x+x^2)*P(n-2,x) - (16*x+4*x^2)*d/dx P(n-2,x) + (12+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 2*x, P(1,x) = 112 + 48*x + 3*x^2.

%t Mathematica

P[0, x_] := 12 + 2*x;
P[1, x_] := 112 + 48*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(32 + 8*x + x^2)*P[n - 2, x] - (16*x + 4*x^2)*D[P[n - 2, x], x] + (12 + 2*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] = 12 + 2*x; v[2] = 112 + 48*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(32 + 8*x + x^2)*v[i - 2] - (16*x + 4*x^2)*deriv(v[i - 2], x) + (12 + 2*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 Stirling2 triangle with scaled diagonals (powers of 4).?
  - 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.