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	x⁴	x⁵	x⁶	x⁷	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
8	12870	48048	41580	10080	350				112928
9	48620	205920	216216	73920	6300				550976
10	184756	875160	1081080	480480	69300	1512			2692288
11	705432	3695120	5250960	2882880	600600	33264			13168256
12	2704156	15519504	24942060	16336320	4504500	432432	6468		64445440
13	10400600	64899744	116396280	88682880	30630600	4324320	168168		315502592
14	40116600	270415600	535422888	465585120	193993800	36756720	2522520	27456	1544840704

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.