A122542 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6\,x^{2} + 6\,x^{3} + n - 9\,n x + 5\,n x^{2} - 5\,n x^{3} - n^{2} + 7\,n^{2} x - 7\,n^{2} x^{2} + n^{2} x^{3} - 2\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(6\,x^{2} - 6\,x^{3} + n + 7\,n x - 5\,n x^{2} + 5\,n x^{3} - n^{2} - n^{2} x + 11\,n^{2} x^{2} - n^{2} x^{3} - 2\,n^{3} x - 6\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x^{2} + 5\,x^{3} - 5\,x^{4} - x^{5} + n x + 6\,n x^{2} + 2\,n x^{3} + 6\,n x^{4} + n x^{5} + 2\,n^{2} x^{2} + 12\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1	1														1
2	1														1
3	1	4													5
4	1	8	6												15
5	1	12	24	8											45
6	1	16	54	48	10										129
7	1	20	96	152	80	12									361
8	1	24	150	352	330	120	14								991
9	1	28	216	680	960	612	168	16							2681
10	1	32	294	1168	2250	2160	1022	224	18						7169
11	1	36	384	1848	4560	6012	4256	1584	288	20					18989
12	1	40	486	2752	8330	14184	13790	7616	2322	360	22				49903
13	1	44	600	3912	14080	29652	37352	28240	12672	3260	440	24			130277
14	1	48	726	5360	22410	56544	88494	86688	53010	19920	4422	528	26		338177
15	1	52	864	7128	34000	100332	189056	230512	182304	92900	29920	5832	624	28	873553

Technical details

Node id: n-3fc33788543ec55d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-cedb3b99895f1c1f

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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: 1;
   3: 1,  4;
   4: 1,  8,   6;
   5: 1, 12,  24,    8;
   6: 1, 16,  54,   48,   10;
   7: 1, 20,  96,  152,   80,   12;
   8: 1, 24, 150,  352,  330,  120,   14;
   9: 1, 28, 216,  680,  960,  612,  168,  16;
  10: 1, 32, 294, 1168, 2250, 2160, 1022, 224, 18;

%F Formula

The row polynomials satisfy -(n^2-n+(2*n^3-7*n^2+9*n)*x-(2*n^3-7*n^2+5*n-6)*x^2-(n^2-5*n+6)*x^3)*P(n,x) = -(n^2-n+(2*n^3+n^2-7*n)*x+(6*n^3-11*n^2+5*n-6)*x^2+(n^2-5*n+6)*x^3)*P(n-1,x) + (n*x+(2*n^2+6*n+1)*x^2+(12*n^2+2*n+5)*x^3+(2*n^2+6*n-5)*x^4+(n-1)*x^5)*d/dx 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[(-(n^2 - n + (2*n^3 + n^2 - 7*n)*x + (6*n^3 - 11*n^2 + 5*n - 6)*x^2 + (n^2 - 5*n + 6)*x^3)*P[n - 1, x] + (n*x + (2*n^2 + 6*n + 1)*x^2 + (12*n^2 + 2*n + 5)*x^3 + (2*n^2 + 6*n - 5)*x^4 + (n - 1)*x^5)*D[P[n - 1, x], x])/(-n^2 + n - (2*n^3 - 7*n^2 + 9*n)*x + (2*n^3 - 7*n^2 + 5*n - 6)*x^2 + (n^2 - 5*n + 6)*x^3)]];
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] = (-(n^2 - n + (2*n^3 + n^2 - 7*n)*x + (6*n^3 - 11*n^2 + 5*n - 6)*x^2 + (n^2 - 5*n + 6)*x^3)*v[i - 1] + (n*x + (2*n^2 + 6*n + 1)*x^2 + (12*n^2 + 2*n + 5)*x^3 + (2*n^2 + 6*n - 5)*x^4 + (n - 1)*x^5)*deriv(v[i - 1], x))/(-n^2 + n - (2*n^3 - 7*n^2 + 9*n)*x + (2*n^3 - 7*n^2 + 5*n - 6)*x^2 + (n^2 - 5*n + 6)*x^3)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 3
by 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 reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

A266213 → 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.