Veronese parent of A109447 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 5\,x + x^{2}\right)\,P_{n-2}(x) + \left(5\,x + 2\,x^{2}\right)\,P_{n-2}'(x) + \left(x^{2}\right)\,P_{n-2}''(x)\)

\(P_{0}(x) = 2 + 4\,x + x^{2}\)

\(P_{1}(x) = 4 + 5\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	2	4	1															7
1	4	5	1															10
2	8	46	46	13	1													114
3	16	65	55	14	1													151
4	32	454	1066	785	225	26	1											2589
5	64	665	1351	910	245	27	1											3263
6	128	4246	20266	29673	17801	4984	679	43	1									77821
7	256	6305	26335	35574	20181	5418	714	44	1									94828
8	512	38854	354106	928465	987505	505302	136017	20010	1596	64	1							2972432
9	1024	58025	465751	1132670	1144165	563409	147147	21120	1650	65	1							3535027
10	2048	352246	5938186	26437433	46117401	38525828	17272123	4438401	675807	61270	3212	89	1					139824045
11	4096	527345	7859215	32566534	54115061	43613856	19012708	4775628	713427	63635	3289	90	1					163254885
12	8192	3180454	97478746	714731745	1956976385	2513729218	1729420693	691662400	169182442	26044018	2547545	156611	5811	118	1			7905124379
13	16384	4766585	129442951	885423630	2314233285	2873141271	1925136213	753655760	181092340	27457430	2650648	161070	5915	119	1			9097183602
14	32768	28665046	1581931306	18748782793	78410677801	149328156432	150345839007	87938535399	31789250493	7395819860	1135041336	116199902	7914998	351680	9725	151	1	526827208698

Technical details

Node id: n-271f55bb02da082d · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-1ecdcc180864d0f3

Observed on the stored rows ?

Degrees: 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16 (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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:    2,     4,      1;
  1:    4,     5,      1;
  2:    8,    46,     46,      13,       1;
  3:   16,    65,     55,      14,       1;
  4:   32,   454,   1066,     785,     225,     26,      1;
  5:   64,   665,   1351,     910,     245,     27,      1;
  6:  128,  4246,  20266,   29673,   17801,   4984,    679,    43,    1;
  7:  256,  6305,  26335,   35574,   20181,   5418,    714,    44,    1;
  8:  512, 38854, 354106,  928465,  987505, 505302, 136017, 20010, 1596, 64, 1;
  9: 1024, 58025, 465751, 1132670, 1144165, 563409, 147147, 21120, 1650, 65, 1;

%F Formula

The row polynomials satisfy P(n,x) = (4+5*x+x^2)*P(n-2,x) + (5*x+2*x^2)*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 2 + 4*x + x^2, P(1,x) = 4 + 5*x + x^2.

%t Mathematica

P[0, x_] := 2 + 4*x + x^2;
P[1, x_] := 4 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 5*x + x^2)*P[n - 2, x] + (5*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 4*x + x^2; v[2] = 4 + 5*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (4 + 5*x + x^2)*v[i - 2] + (5*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), 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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from veronese parent of Binomial coefficients C(n,k) with n-k odd, read by rows.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Parent-row provers of polytool 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.