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	x⁵	x⁶	x⁷	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
8	40320	2903040	15240960	13547520	1814400				33546240
9	362880	32659200	228614400	304819200	81648000	2177280			650280960
10	3628800	399168000	3592512000	6706022400	2993760000	239500800			13934592000
11	39916800	5269017600	59276448000	147532492800	98794080000	15807052800	279417600		326998425600
12	479001600	74724249600	1027458432000	3287866982400	3082375296000	821966745600	43589145600		8338459852800
13	6227020800	1133317785600	18699743462400	74798973849600	93498717312000	37399486924800	3966612249600	49816166400	229552894771200
14	87178291200	18307441152000	356995102464000	1745309389824000	2804961519360000	1570778450841600	277662857472000	10461394944000	6784563334348800

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.