Veronese parent of A098172 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 - 3\,x + 2\,n + 3\,n x\right)\,P_{n}(x) = \left(x^{2} + x^{3} + 2\,n x^{2} + 5\,n x^{3} + 3\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	x²⁹	x³⁰	x³¹	x³²	x³³	x³⁴	x³⁵	x³⁶	x³⁷	x³⁸	x³⁹	x⁴⁰	x⁴¹	x⁴²	x⁴³	x⁴⁴	P_n(1)
1	3																																											3
2	0	0	5	6																																								11
3	0	0	0	0	7	16	9																																					32
4	0	0	0	0	0	0	9	30	33	12																																		84
5	0	0	0	0	0	0	0	0	11	48	78	56	15																															208
6	0	0	0	0	0	0	0	0	0	0	13	70	150	160	85	18																												496
7	0	0	0	0	0	0	0	0	0	0	0	0	15	96	255	360	285	120	21																									1152
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	17	126	399	700	735	462	161	24																						2624
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	19	160	588	1232	1610	1344	700	208	27																			5888
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	21	198	828	2016	3150	3276	2268	1008	261	30																13056
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	23	240	1125	3120	5670	7056	6090	3600	1395	320	33													28672
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	25	286	1485	4620	9570	13860	14322	10560	5445	1870	385	36										62464
13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	27	336	1914	6600	15345	25344	30492	26928	17325	7920	2442	456	39							135168
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	29	390	2418	9152	23595	43758	60060	61776	47619	27170	11154	3120	533	42				290816
15	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	31	448	3003	12376	35035	72072	111111	130416	117117	80080	41041	15288	3913	616	45	622592

Technical details

Node id: n-8fec94b874cdd565 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-76e137d3605ff796

Observed on the stored rows ?

Degrees: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44 (affine: slope 3, 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 0, 0, 3;
   2: 0, 0, 0, 0, 5, 6;
   3: 0, 0, 0, 0, 0, 0, 7, 16, 9;
   4: 0, 0, 0, 0, 0, 0, 0,  0, 9, 30, 33, 12;
   5: 0, 0, 0, 0, 0, 0, 0,  0, 0,  0, 11, 48, 78, 56,  15;
   6: 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 13, 70, 150, 160,  85,  18;
   7: 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  15,  96, 255, 360, 285, 120,  21;
   8: 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  17, 126, 399, 700, 735,  462,  161,   24;
   9: 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  19, 160, 588, 1232, 1610, 1344,  700,  208,   27;
  10: 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  21,  198,  828, 2016, 3150, 3276, 2268, 1008, 261, 30;

%F Formula

The row polynomials satisfy (2*n-1+(3*n-3)*x)*P(n,x) = ((2*n+1)*x^2+(5*n+1)*x^3+3*n*x^4)*P(n-1,x) for n >= 2, with P(1,x) = 3*x^2.

%t Mathematica

P[1, x_] := 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n + 1)*x^2 + (5*n + 1)*x^3 + 3*n*x^4)*P[n - 1, x])/(2*n - 1 + (3*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 3*x^2; for(n = 2, N, my(i = n); v[i] = (((2*n + 1)*x^2 + (5*n + 1)*x^3 + 3*n*x^4)*v[i - 1])/(2*n - 1 + (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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
  - 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.