Veronese parent of A098172 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	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	P_n(1)
1	1	1																																											2
2	0	0	3	5	1																																								9
3	0	0	0	0	5	13	9	1																																					28
4	0	0	0	0	0	0	7	25	30	13	1																																		76
5	0	0	0	0	0	0	0	0	9	41	70	54	17	1																															192
6	0	0	0	0	0	0	0	0	0	0	11	61	135	150	85	21	1																												464
7	0	0	0	0	0	0	0	0	0	0	0	0	13	85	231	335	275	123	25	1																									1088
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	15	113	364	651	700	455	168	29	1																						2496
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	17	145	540	1148	1526	1302	700	220	33	1																			5632
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	19	181	765	1884	2982	3150	2226	1020	279	37	1																12544
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	21	221	1045	2925	5370	6762	5922	3570	1425	345	41	1													27648
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	23	265	1386	4345	9075	13266	13860	10362	5445	1925	418	45	1										60416
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	25	313	1794	6226	14575	24255	29436	26268	17127	7975	2530	498	49	1							131072
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	27	365	2275	8658	22451	41899	57915	60060	46761	27027	11297	3250	585	53	1				282624
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	29	421	2835	11739	33397	69069	107107	126555	114543	79079	41041	15561	4095	679	57	1	606208

Technical details

Node id: n-220c6973448e8e72 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-19afff2b02650d12

Observed on the stored rows ?

Degrees: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43 (affine: slope 3, intercept -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1, 1;
   2: 0, 0, 3, 5, 1;
   3: 0, 0, 0, 0, 5, 13, 9,  1;
   4: 0, 0, 0, 0, 0,  0, 7, 25, 30, 13,  1;
   5: 0, 0, 0, 0, 0,  0, 0,  0,  9, 41, 70, 54,  17,   1;
   6: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 11, 61, 135, 150,  85,  21,   1;
   7: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  13,  85, 231, 335, 275, 123,  25,    1;
   8: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  15, 113, 364, 651, 700,  455,  168,   29,    1;
   9: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  17, 145, 540, 1148, 1526, 1302,  700,  220,   33,    1;
  10: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  19,  181,  765, 1884, 2982, 3150, 2226, 1020, 279, 37, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = (((2*n - 1)*x^2 + (5*n - 2)*x^3 + 3*n*x^4 + x^5)*v[i - 1])/(2*n - 3 + (3*n - 4)*x + x^2)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.