A159854 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 3\,x + n + 3\,n x\right)\,P_{n}(x) = \left(8 + 24\,x + 18\,x^{2} + 2\,n + 9\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 6 + 18\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	6	18															24
1	15	72	81														168
2	36	234	486	324													1080
3	84	684	2025	2592	1215												6600
4	192	1872	7128	13284	12150	4374											39000
5	432	4896	22680	55080	74115	52488	15309										225000
6	960	12384	67392	200880	354780	371790	214326	52488									1275000
7	2112	30528	190512	671328	1462860	2020788	1729917	839808	177147								7125000
8	4608	73728	518400	2104704	5443200	9307872	10532592	7610760	3188646	590490							39375000
9	9984	175104	1368576	6283008	18779040	38211264	53642736	51333264	32063607	11809800	1948617						215625000
10	21504	410112	3525120	18040320	61119360	144027072	241024896	286584480	237376980	130498290	42869574	6377292					1171875000
11	46080	949248	8895744	50181120	189889920	508083840	986144544	1399540032	1441976580	1052253180	516383505	153055008	20726199				6328125000
12	98304	2174976	22063104	135945216	567959040	1699771392	3750582528	6180147072	7607400768	6911094960	4505202504	1996092396	538881174	66961566			33984375000
13	208896	4939776	53913600	360142848	1645712640	5444195328	13450364928	25219434240	36072797904	39180192480	31817018376	18736483896	7565062635	1874923848	215233605		181640625000
14	442368	11132928	130056192	935940096	4641960960	16812956160	45955413504	96554405376	157261186368	198603044640	192928671936	141601391568	76023697932	28190819286	6457008150	688747536	966796875000

Technical details

Node id: n-eee9ba3e57c20bfc · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-e4e69340af65702e

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    6,     18;
  1:   15,     72,      81;
  2:   36,    234,     486,     324;
  3:   84,    684,    2025,    2592,     1215;
  4:  192,   1872,    7128,   13284,    12150,     4374;
  5:  432,   4896,   22680,   55080,    74115,    52488,    15309;
  6:  960,  12384,   67392,  200880,   354780,   371790,   214326,    52488;
  7: 2112,  30528,  190512,  671328,  1462860,  2020788,  1729917,   839808,   177147;
  8: 4608,  73728,  518400, 2104704,  5443200,  9307872, 10532592,  7610760,  3188646,   590490;
  9: 9984, 175104, 1368576, 6283008, 18779040, 38211264, 53642736, 51333264, 32063607, 11809800, 1948617;

%F Formula

The row polynomials satisfy (n+3+(3*n+3)*x)*P(n,x) = (2*n+8+(9*n+24)*x+(9*n+18)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 6 + 18*x.

%t Mathematica

P[0, x_] := 6 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (9*n + 24)*x + (9*n + 18)*x^2)*P[n - 1, x])/(n + 3 + (3*n + 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 6 + 18*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (9*n + 24)*x + (9*n + 18)*x^2)*v[i - 1])/(n + 3 + (3*n + 3)*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 basis_map_A013610 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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.