Veronese parent of A098172 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	P_n(1)
1	1																						1
2	0	2	4	1																			7
3	0	0	0	4	11	8	1																24
4	0	0	0	0	0	6	22	27	12	1													68
5	0	0	0	0	0	0	0	8	37	64	50	16	1										176
6	0	0	0	0	0	0	0	0	0	10	56	125	140	80	20	1							432
7	0	0	0	0	0	0	0	0	0	0	0	12	79	216	315	260	117	24	1				1024
8	0	0	0	0	0	0	0	0	0	0	0	0	0	14	106	343	616	665	434	161	28	1	2368

All 15 rows

Technical details

Node id: n-6332792258acccee · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e12c7efa2d1f8675

Observed on the stored rows ?

Degrees: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42 (affine: slope 3, intercept -3)
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;
   2: 0, 2, 4, 1;
   3: 0, 0, 0, 4, 11, 8,  1;
   4: 0, 0, 0, 0,  0, 6, 22, 27, 12,  1;
   5: 0, 0, 0, 0,  0, 0,  0,  8, 37, 64, 50,  16,   1;
   6: 0, 0, 0, 0,  0, 0,  0,  0,  0, 10, 56, 125, 140,  80,  20,   1;
   7: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  12,  79, 216, 315, 260, 117,  24,    1;
   8: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  14, 106, 343, 616, 665,  434,  161,   28,    1;
   9: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  16, 137, 512, 1092, 1456, 1246,  672,  212,   32,   1;
  10: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  18,  172,  729, 1800, 2856, 3024, 2142, 984, 270, 36, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 2)*x^2 + (5*n - 4)*x^3 + (3*n - 1)*x^4 + x^5)*P[n - 1, x])/(2*n - 4 + (3*n - 5)*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; for(n = 2, N, my(i = n); v[i] = (((2*n - 2)*x^2 + (5*n - 4)*x^3 + (3*n - 1)*x^4 + x^5)*v[i - 1])/(2*n - 4 + (3*n - 5)*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).?
  - 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
  - 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.