A115951 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 3\,n^{2} + 2\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} + 2\,n + 14\,n x + 4\,n x^{2} - 4\,n^{2} + 2\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x + 16\,x^{2} + 8\,x^{3} + 12\,n x + 20\,n x^{2} + 8\,n x^{3} + 4\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	4							4
2	0	4	18						22
3	0	0	36	80					116
4	0	0	18	240	350				608
5	0	0	0	240	1400	1512			3152
6	0	0	0	80	2100	7560	6468		16208
7	0	0	0	0	1400	15120	38808	27456	82784

All 15 rows

Technical details

Node id: n-c58260973df006a6 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-466ef85da4d3295f

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, 4, 18;
  3: 0, 0, 36,  80;
  4: 0, 0, 18, 240,  350;
  5: 0, 0,  0, 240, 1400,  1512;
  6: 0, 0,  0,  80, 2100,  7560,   6468;
  7: 0, 0,  0,   0, 1400, 15120,  38808,  27456;
  8: 0, 0,  0,   0,  350, 15120,  97020, 192192, 115830;
  9: 0, 0,  0,   0,    0,  7560, 129360, 576576, 926640, 486200;

%F Formula

The row polynomials satisfy (n^3+3*n^2+2*n+2*n^2*x)*P(n,x) = -(2*n^3+4*n^2-2*n-4-(2*n^2+14*n+8)*x-(4*n+4)*x^2)*P(n-1,x) + ((4*n^2+12*n+8)*x+(4*n^2+20*n+16)*x^2+(8*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[(-(2*n^3 + 4*n^2 - 2*n - 4 - (2*n^2 + 14*n + 8)*x - (4*n + 4)*x^2)*P[n - 1, x] + ((4*n^2 + 12*n + 8)*x + (4*n^2 + 20*n + 16)*x^2 + (8*n + 8)*x^3)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + 2*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^3 + 4*n^2 - 2*n - 4 - (2*n^2 + 14*n + 8)*x - (4*n + 4)*x^2)*v[i - 1] + ((4*n^2 + 12*n + 8)*x + (4*n^2 + 20*n + 16)*x^2 + (8*n + 8)*x^3)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + 2*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 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.