A147644 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-6 - 9\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

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

\(P_{2}(x) = 4 + 12\,x + 9\,x^{2}\)

\(P_{3}(x) = 12 + 54\,x + 72\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	4	12	9													25
3	12	54	72	27												165
4	24	144	306	270	81											825
5	48	360	1044	1458	972	243										4125
6	96	864	3168	6048	6318	3402	729									20625
7	192	2016	8928	21600	30780	25758	11664	2187								103125
8	384	4608	23904	69984	126360	143856	100602	39366	6561							515625
9	768	10368	61632	211680	462672	666792	632772	380538	131220	19683						2578125
10	1536	23040	154368	608256	1560384	2721600	3265920	2659392	1404054	433026	59049					12890625
11	3072	50688	377856	1679616	4945536	10124352	14696640	15116544	10786284	5078214	1417176	177147				64453125
12	6144	110592	907776	4492800	14929920	35085312	59766336	74323008	66922200	42515280	18068994	4605822	531441			322265625
13	12288	239616	2147328	11708928	43338240	114960384	224788608	327945024	356813424	285797160	163683828	63418626	14880348	1594323		1611328125
14	24576	516096	5013504	29859840	121803264	359935488	794458368	1330255872	1697461920	1642034592	1184759136	617888736	220016574	47829690	4782969	8056640625

Technical details

Node id: n-3a40c713155476e9 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-13ad852c44bed6fa

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:   1;
  1:   2,     3;
  2:   4,    12,     9;
  3:  12,    54,    72,     27;
  4:  24,   144,   306,    270,     81;
  5:  48,   360,  1044,   1458,    972,    243;
  6:  96,   864,  3168,   6048,   6318,   3402,    729;
  7: 192,  2016,  8928,  21600,  30780,  25758,  11664,   2187;
  8: 384,  4608, 23904,  69984, 126360, 143856, 100602,  39366,   6561;
  9: 768, 10368, 61632, 211680, 462672, 666792, 632772, 380538, 131220, 19683;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (2*n-6+(3*n-9)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 4 + 12*x + 9*x^2, P(3,x) = 12 + 54*x + 72*x^2 + 27*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 4 + 12*x + 9*x^2;
P[3, x_] := 12 + 54*x + 72*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 6 + (3*n - 9)*x)*P[n - 1, x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; v[3] = 4 + 12*x + 9*x^2; v[4] = 12 + 54*x + 72*x^2 + 27*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2*n - 6 + (3*n - 9)*x)*v[i - 1])/(n - 3)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 4
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n, k) = binomial(n, k) if n < 3, otherwise T(n, k) = binomial(n, k) + 2*binomial(n-2, k-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.