basis-map coefficients of A371080, shifted → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 20\,x - 4\,x^{2} + 30\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 4\,x + 6\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	10	8													18
3	78	96	24												198
4	836	1272	528	64											2700
5	11428	19880	10680	2240	160										44388
6	190128	363792	230520	64320	8160	384									857304
7	3728912	7672560	5471256	1836800	315840	26880	896								19053144
8	84249184	183711968	143385312	54999616	11540480	1349376	82432	2048							479320416
9	2154918240	4929795936	4135768704	1757351232	426011040	61431552	5209344	239616	4608						13470730272
10	61554926720	146653959168	130590911520	60234551040	16321687200	2742509952	289201920	18616320	668160	10240					418407042240
11	1942284776320	4792492441088	4488280134624	2215513512960	656558680800	124189857792	15364449408	1238138880	62599680	1802240	22528				14235986416320
12	67094225882880	170713882102272	166979001900480	87296206466304	27861778929600	5798195998848	812167614720	77215463424	4914201600	200386560	4730880	49152			526637793726720
13	2518408836258560	6584790541315584	6690364716624576	3675236977654528	1249126336560960	281401521388416	43579325741952	4713355266048	355560251904	18334750720	615892992	12140544	106496		21047996133953280
14	102065485158246400	273465256997657600	287368960288929792	164850451683846656	59153145906574720	14253964063282944	2399776579584384	288144098052096	24827459424768	1523523921920	64972171264	1829388288	30564352	229376	903871602591874560

Technical details

Node id: n-54c25c7a8c92a98e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d905b37198b1e6f5

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:          2;
  2:         10,          8;
  3:         78,         96,         24;
  4:        836,       1272,        528,         64;
  5:      11428,      19880,      10680,       2240,       160;
  6:     190128,     363792,     230520,      64320,      8160,      384;
  7:    3728912,    7672560,    5471256,    1836800,    315840,    26880,     896;
  8:   84249184,  183711968,  143385312,   54999616,  11540480,  1349376,   82432,   2048;
  9: 2154918240, 4929795936, 4135768704, 1757351232, 426011040, 61431552, 5209344, 239616, 4608;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(12*n-20)*x+4*x^2)*P(n-2,x) + (6*n-7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*P[n - 2, x] + (6*n - 7 + 4*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*v[i - 2] + (6*n - 7 + 4*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
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 basis_map_A013609 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    ↑ Parent-row provers of polytool (above)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.