Veronese parent of A109447 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + 2\,n x + 3\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(4\,x + 14\,x^{2} + 16\,x^{3} + 6\,x^{4} - 2\,n - 5\,n x + 8\,n x^{2} + 22\,n x^{3} + 11\,n x^{4} - 3\,n^{2} x - 3\,n^{2} x^{2} + 6\,n^{2} x^{3} + 6\,n^{2} x^{4} - n^{3} x^{2} + n^{3} x^{4}\right)\,P_{n-1}(x) + \left(1 + 4\,x + x^{2} - 12\,x^{3} - 16\,x^{4} - 6\,x^{5} + 2\,n x + 5\,n x^{2} - n x^{3} - 9\,n x^{4} - 5\,n x^{5} + n^{2} x^{2} + n^{2} x^{3} - n^{2} x^{4} - n^{2} x^{5}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	0	2	3															5
1	1	4	3															8
2	0	2	9	12	5													28
3	1	8	18	16	5													48
4	0	2	15	40	50	30	7											144
5	1	12	45	80	75	36	7											256
6	0	2	21	84	175	210	147	56	9									704
7	1	16	84	224	350	336	196	64	9									1280
8	0	2	27	144	420	756	882	672	324	90	11							3328
9	1	20	135	480	1050	1512	1470	960	405	100	11							6144
10	0	2	33	220	825	1980	3234	3696	2970	1650	605	132	13					15360
11	1	24	198	880	2475	4752	6468	6336	4455	2200	726	144	13					28672
12	0	2	39	312	1430	4290	9009	13728	15444	12870	7865	3432	1014	182	15			69632
13	1	28	273	1456	5005	12012	21021	27456	27027	20020	11011	4368	1183	196	15			131072
14	0	2	45	420	2275	8190	21021	40040	57915	64350	55055	36036	17745	6370	1575	240	17	311296

Technical details

Node id: n-2895517eccddbe67 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-c02dd46908d39d3e

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 0,  2,   3;
  1: 1,  4,   3;
  2: 0,  2,   9,  12,    5;
  3: 1,  8,  18,  16,    5;
  4: 0,  2,  15,  40,   50,   30,    7;
  5: 1, 12,  45,  80,   75,   36,    7;
  6: 0,  2,  21,  84,  175,  210,  147,  56,   9;
  7: 1, 16,  84, 224,  350,  336,  196,  64,   9;
  8: 0,  2,  27, 144,  420,  756,  882, 672, 324,  90, 11;
  9: 1, 20, 135, 480, 1050, 1512, 1470, 960, 405, 100, 11;

%F Formula

The row polynomials satisfy (2+(2*n+4)*x+(n^2+3*n+2)*x^2)*P(n,x) = -(2*n+(3*n^2+5*n-4)*x+(n^3+3*n^2-8*n-14)*x^2-(6*n^2+22*n+16)*x^3-(n^3+6*n^2+11*n+6)*x^4)*P(n-1,x) + (1+(2*n+4)*x+(n^2+5*n+1)*x^2+(n^2-n-12)*x^3-(n^2+9*n+16)*x^4-(n^2+5*n+6)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2*x + 3*x^2.

%t Mathematica

P[0, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (3*n^2 + 5*n - 4)*x + (n^3 + 3*n^2 - 8*n - 14)*x^2 - (6*n^2 + 22*n + 16)*x^3 - (n^3 + 6*n^2 + 11*n + 6)*x^4)*P[n - 1, x] + (1 + (2*n + 4)*x + (n^2 + 5*n + 1)*x^2 + (n^2 - n - 12)*x^3 - (n^2 + 9*n + 16)*x^4 - (n^2 + 5*n + 6)*x^5)*D[P[n - 1, x], x])/(2 + (2*n + 4)*x + (n^2 + 3*n + 2)*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*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n + (3*n^2 + 5*n - 4)*x + (n^3 + 3*n^2 - 8*n - 14)*x^2 - (6*n^2 + 22*n + 16)*x^3 - (n^3 + 6*n^2 + 11*n + 6)*x^4)*v[i - 1] + (1 + (2*n + 4)*x + (n^2 + 5*n + 1)*x^2 + (n^2 - n - 12)*x^3 - (n^2 + 9*n + 16)*x^4 - (n^2 + 5*n + 6)*x^5)*deriv(v[i - 1], x))/(2 + (2*n + 4)*x + (n^2 + 3*n + 2)*x^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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.