A228955 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-6 + 2\,x + 3\,n x + n^{2} x\right)\,P_{n}(x) = \left(-6\,n - 10\,n x + 6\,n x^{2} + 3\,n^{2} x + 5\,n^{2} x^{2} + n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-12\,x + 24\,x^{2} - 12\,x^{3} + 10\,n x^{2} - 10\,n x^{3} + 2\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	1					1
1	1	2				3
2	2	12				14
3	6	72	18			96
4	24	480	360			864
5	120	3600	5400	480		9600
6	720	30240	75600	20160		126720
7	5040	282240	1058400	564480	25200	1935360

All 15 rows

Technical details

Node id: n-3a0d63362129dc11 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9ad63b3855519f4d

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
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:      1,        2;
  2:      2,       12;
  3:      6,       72,        18;
  4:     24,      480,       360;
  5:    120,     3600,      5400,       480;
  6:    720,    30240,     75600,     20160;
  7:   5040,   282240,   1058400,    564480,    25200;
  8:  40320,  2903040,  15240960,  13547520,  1814400;
  9: 362880, 32659200, 228614400, 304819200, 81648000, 2177280;

%F Formula

The row polynomials satisfy -(6-(n^2+3*n+2)*x)*P(n,x) = -(6*n-(n^3+3*n^2-10*n)*x-(n^3+5*n^2+6*n)*x^2)*P(n-1,x) - (12*x-(2*n^2+10*n+24)*x^2+(2*n^2+10*n+12)*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[(-(6*n - (n^3 + 3*n^2 - 10*n)*x - (n^3 + 5*n^2 + 6*n)*x^2)*P[n - 1, x] - (12*x - (2*n^2 + 10*n + 24)*x^2 + (2*n^2 + 10*n + 12)*x^3)*D[P[n - 1, x], x])/(-6 + (n^2 + 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] = (-(6*n - (n^3 + 3*n^2 - 10*n)*x - (n^3 + 5*n^2 + 6*n)*x^2)*v[i - 1] - (12*x - (2*n^2 + 10*n + 24)*x^2 + (2*n^2 + 10*n + 12)*x^3)*deriv(v[i - 1], x))/(-6 + (n^2 + 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 Table: T(n,k) = n!*binomial(n+1,2*k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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.