Veronese parent of A098172 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	P_n(1)
1	2	1																						3
2	0	0	4	6	1																			11
3	0	0	0	0	6	15	10	1																32
4	0	0	0	0	0	0	8	28	33	14	1													84
5	0	0	0	0	0	0	0	0	10	45	76	58	18	1										208
6	0	0	0	0	0	0	0	0	0	0	12	66	145	160	90	22	1							496
7	0	0	0	0	0	0	0	0	0	0	0	0	14	91	246	355	290	129	26	1				1152
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	16	120	385	686	735	476	175	30	1	2624

All 15 rows

Technical details

Node id: n-d669dc3501691c45 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f15ecd8da6d2c93b

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 0, 2, 1;
   2: 0, 0, 0, 4, 6, 1;
   3: 0, 0, 0, 0, 0, 6, 15, 10,  1;
   4: 0, 0, 0, 0, 0, 0,  0,  8, 28, 33, 14,  1;
   5: 0, 0, 0, 0, 0, 0,  0,  0,  0, 10, 45, 76, 58,  18,   1;
   6: 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0, 12, 66, 145, 160,  90,  22,   1;
   7: 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  14,  91, 246, 355, 290, 129,  26,    1;
   8: 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,   0,   0,  16, 120, 385, 686, 735,  476,  175,   30,    1;
   9: 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  18, 153, 568, 1204, 1596, 1358,  728,  228,   34,    1;
  10: 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  20,  190,  801, 1968, 3108, 3276, 2310, 1056, 288, 38, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

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