A008556 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2} + n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(-4 - 8\,x - 4\,x^{2} - 10\,n x - 2\,n x^{2} + 16\,n^{2} + 18\,n^{2} x + 4\,n^{2} x^{2} + 4\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 16\,x^{2} - 8\,x^{3} - 16\,n x - 28\,n x^{2} - 12\,n x^{3} - 4\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	6	4			10
3	20	24			44
4	70	120	18		208
5	252	560	180		992
6	924	2520	1260	80	4784
7	3432	11088	7560	1120	23200

All 15 rows

Technical details

Node id: n-163c88df01802142 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-32cde9592462b285

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:     2;
  2:     6,      4;
  3:    20,     24;
  4:    70,    120,     18;
  5:   252,    560,    180;
  6:   924,   2520,   1260,    80;
  7:  3432,  11088,   7560,  1120;
  8: 12870,  48048,  41580, 10080,  350;
  9: 48620, 205920, 216216, 73920, 6300;

%F Formula

The row polynomials satisfy (4*n^2+2*n+(n^3+n^2)*x)*P(n,x) = (16*n^2-4+(4*n^3+18*n^2-10*n-8)*x+(2*n^3+4*n^2-2*n-4)*x^2)*P(n-1,x) - ((16*n+8)*x+(4*n^2+28*n+16)*x^2+(4*n^2+12*n+8)*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[((16*n^2 - 4 + (4*n^3 + 18*n^2 - 10*n - 8)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - ((16*n + 8)*x + (4*n^2 + 28*n + 16)*x^2 + (4*n^2 + 12*n + 8)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 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] = ((16*n^2 - 4 + (4*n^3 + 18*n^2 - 10*n - 8)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*v[i - 1] - ((16*n + 8)*x + (4*n^2 + 28*n + 16)*x^2 + (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 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 of coefficients of Legendre polynomials 2^n P_n (x).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
    
    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.