A107131 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 3\,n + n x - n^{2} + n^{2} x\right)\,P_{n}(x) = \left(-4 - 5\,n x + n x^{2} + 4\,n^{2} - 7\,n^{2} x + 3\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x + 6\,x^{2} - x^{3} - 8\,n x + 10\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	2	3													5
3	0	0	9	4												13
4	0	0	6	24	5											35
5	0	0	0	40	50	6										96
6	0	0	0	20	150	90	7									267
7	0	0	0	0	175	420	147	8								750
8	0	0	0	0	70	840	980	224	9							2123
9	0	0	0	0	0	756	2940	2016	324	10						6046
10	0	0	0	0	0	252	4410	8400	3780	450	11					17303
11	0	0	0	0	0	0	3234	18480	20790	6600	605	12				49721
12	0	0	0	0	0	0	924	22176	62370	46200	10890	792	13			143365
13	0	0	0	0	0	0	0	13728	108108	180180	94380	17160	1014	14		414584
14	0	0	0	0	0	0	0	3432	108108	420420	462462	180180	26026	1274	15	1201917

Technical details

Node id: n-716c08682c9e63d1 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-5192a5fe8d8bfcb0

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, 2;
  2: 0, 2, 3;
  3: 0, 0, 9,  4;
  4: 0, 0, 6, 24,   5;
  5: 0, 0, 0, 40,  50,   6;
  6: 0, 0, 0, 20, 150,  90,    7;
  7: 0, 0, 0,  0, 175, 420,  147,    8;
  8: 0, 0, 0,  0,  70, 840,  980,  224,   9;
  9: 0, 0, 0,  0,   0, 756, 2940, 2016, 324, 10;

%F Formula

The row polynomials satisfy -(n^2+3*n+2-(n^2+n)*x)*P(n,x) = (4*n^2-4-(7*n^2+5*n)*x+(3*n^2+n)*x^2)*P(n-1,x) - ((8*n+8)*x-(10*n+6)*x^2+(2*n+1)*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[((4*n^2 - 4 - (7*n^2 + 5*n)*x + (3*n^2 + n)*x^2)*P[n - 1, x] - ((8*n + 8)*x - (10*n + 6)*x^2 + (2*n + 1)*x^3)*D[P[n - 1, x], x])/(-n^2 - 3*n - 2 + (n^2 + n)*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] = ((4*n^2 - 4 - (7*n^2 + 5*n)*x + (3*n^2 + n)*x^2)*v[i - 1] - ((8*n + 8)*x - (10*n + 6)*x^2 + (2*n + 1)*x^3)*deriv(v[i - 1], x))/(-n^2 - 3*n - 2 + (n^2 + n)*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 A Motzkin related triangle.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.