A133081 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

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

\(P_{2}(x) = 2 + 12\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1															1
2	2	12	16													30
3	1	8	16													25
4	2	28	144	320	256											750
5	1	16	96	256	256											625
6	2	44	400	1920	5120	7168	4096									18750
7	1	24	240	1280	3840	6144	4096									15625
8	2	60	784	5824	26880	78848	143360	147456	65536							468750
9	1	32	448	3584	17920	57344	114688	131072	65536							390625
10	2	76	1296	13056	86016	387072	1204224	2555904	3538944	2883584	1048576					11718750
11	1	40	720	7680	53760	258048	860160	1966080	2949120	2621440	1048576					9765625
12	2	92	1936	24640	211200	1284096	5677056	18382848	43253760	72089600	80740352	54525952	16777216			292968750
13	1	48	1056	14080	126720	811008	3784704	12976128	32440320	57671680	69206016	50331648	16777216			244140625
14	2	108	2704	41600	439296	3367936	19329024	84344832	281149440	712245248	1349517312	1853882368	1744830464	1006632960	268435456	7324218750

Technical details

Node id: n-4270c66f3fd5c398 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-06d77c36a30b826d

Observed on the stored rows ?

Degrees: 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14 (irregular)
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: 1;
  1: 1;
  2: 2, 12,  16;
  3: 1,  8,  16;
  4: 2, 28, 144,  320,   256;
  5: 1, 16,  96,  256,   256;
  6: 2, 44, 400, 1920,  5120,  7168,   4096;
  7: 1, 24, 240, 1280,  3840,  6144,   4096;
  8: 2, 60, 784, 5824, 26880, 78848, 143360, 147456, 65536;
  9: 1, 32, 448, 3584, 17920, 57344, 114688, 131072, 65536;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 2 + 12*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (8*n - 16)*x + (16*n - 32)*x^2)*P[n - 2, x])/(n - 2)]];
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] = 1; v[3] = 2 + 12*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (8*n - 16)*x + (16*n - 32)*x^2)*v[i - 2])/(n - 2)); 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from An interpolation operator, companion to A133080.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.