A386789 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(2 - 2\,n - 7\,n x - 2\,n x^{2} - 2\,n^{2} x + n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 6\,x^{2} - 2\,x^{3} + 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	4							4
2	0	6	18						24
3	0	8	60	80					148
4	0	10	135	420	350				915
5	0	12	252	1344	2520	1512			5640
6	0	14	420	3360	10500	13860	6468		34622
7	0	16	648	7200	33000	71280	72072	27456	211672

All 15 rows

Technical details

Node id: n-0bb4961a16b9a9e4 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7844ae331c0f03a8

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 0,  4;
  2: 0,  6,   18;
  3: 0,  8,   60,    80;
  4: 0, 10,  135,   420,    350;
  5: 0, 12,  252,  1344,   2520,   1512;
  6: 0, 14,  420,  3360,  10500,  13860,    6468;
  7: 0, 16,  648,  7200,  33000,  71280,   72072,   27456;
  8: 0, 18,  945, 13860,  86625, 270270,  441441,  360360,  115830;
  9: 0, 20, 1320, 24640, 200200, 840840, 1961960, 2562560, 1750320, 486200;

%F Formula

The row polynomials satisfy -(2*n-(n^3-n^2)*x)*P(n,x) = -(2*n-2-(n^3-2*n^2-7*n)*x-(2*n^3-2*n)*x^2)*P(n-1,x) - (4*x-(2*n^2-6)*x^2-(2*n^2-2)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 2 - (n^3 - 2*n^2 - 7*n)*x - (2*n^3 - 2*n)*x^2)*P[n - 1, x] - (4*x - (2*n^2 - 6)*x^2 - (2*n^2 - 2)*x^3)*D[P[n - 1, x], x])/(-2*n + (n^3 - n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 2 - (n^3 - 2*n^2 - 7*n)*x - (2*n^3 - 2*n)*x^2)*v[i - 1] - (4*x - (2*n^2 - 6)*x^2 - (2*n^2 - 2)*x^3)*deriv(v[i - 1], x))/(-2*n + (n^3 - n^2)*x)); 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 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] 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

Related ?

Same fingerprint ?

A357367 → θ+1

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.