A079510 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9\,x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-4}(x) + \left(-4\,x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-9\,x^{2} + 9\,n x^{2} + 18\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-4\,x^{3} + 4\,n x^{3} + 8\,n^{2} x^{3}\right)\,P_{n-3}'(x) + \left(4\,x + 4\,n x + 9\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} - 4\,n x^{2} + 4\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 8\,x\)

\(P_{2}(x) = 12\,x + 54\,x^{2}\)

\(P_{3}(x) = 180\,x^{2} + 384\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	8														8
2	0	12	54													66
3	0	0	180	384												564
4	0	0	135	2080	3000											5215
5	0	0	0	3360	23100	25920										52380
6	0	0	0	1680	59500	263088	246960									571228
7	0	0	0	0	63000	951552	3146976	2580480								6742008
8	0	0	0	0	23625	1587600	14879340	39896064	29393280							85779909
9	0	0	0	0	0	1247400	34560680	235445760	537593760	362880000						1171727600
10	0	0	0	0	0	374220	42446250	710321920	3833274456	7699824000	4829932800					17116173646
11	0	0	0	0	0	0	26486460	1206965760	14363846640	64753603200	117070462080	68976230400				266397594540
12	0	0	0	0	0	0	6621615	1170449280	31545373860	292052560800	1139692409856	1885930076160	1052366515200			4402764006771
13	0	0	0	0	0	0	0	605404800	42124822740	793492700000	6044296418160	20939123552256	32120196675840	17086945075200		77026784648996
14	0	0	0	0	0	0	0	129729600	33713479800	1364262900000	19702646483520	128243106351360	401816048250240	577084827110400	294226732800000	1422471467104920

Technical details

Node id: n-2c096deb92ac6bc1 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-d5c7a04bc225be4d

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,  8;
  2: 0, 12,  54;
  3: 0,  0, 180,  384;
  4: 0,  0, 135, 2080,  3000;
  5: 0,  0,   0, 3360, 23100,   25920;
  6: 0,  0,   0, 1680, 59500,  263088,   246960;
  7: 0,  0,   0,    0, 63000,  951552,  3146976,   2580480;
  8: 0,  0,   0,    0, 23625, 1587600, 14879340,  39896064,  29393280;
  9: 0,  0,   0,    0,     0, 1247400, 34560680, 235445760, 537593760, 362880000;

%F Formula

The row polynomials satisfy P(n,x) = (9*n^2-9)*x^2*P(n-4,x) + (4*n^2-4)*x^3*d/dx P(n-4,x) + (18*n^2+9*n-9)*x^2*P(n-3,x) + (8*n^2+4*n-4)*x^3*d/dx P(n-3,x) + ((4*n+4)*x+(9*n^2+9*n)*x^2)*P(n-2,x) - ((4*n+4)*x^2-(4*n^2+4*n)*x^3)*d/dx P(n-2,x) + (4*n+4)*x*P(n-1,x) - (4*n+4)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 8*x, P(2,x) = 12*x + 54*x^2, P(3,x) = 180*x^2 + 384*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 8*x;
P[2, x_] := 12*x + 54*x^2;
P[3, x_] := 180*x^2 + 384*x^3;
P[n_, x_] := P[n, x] = Expand[(9*n^2 - 9)*x^2*P[n - 4, x] + (4*n^2 - 4)*x^3*D[P[n - 4, x], x] + (18*n^2 + 9*n - 9)*x^2*P[n - 3, x] + (8*n^2 + 4*n - 4)*x^3*D[P[n - 3, x], x] + ((4*n + 4)*x + (9*n^2 + 9*n)*x^2)*P[n - 2, x] - ((4*n + 4)*x^2 - (4*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (4*n + 4)*x*P[n - 1, x] - (4*n + 4)*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8*x; v[3] = 12*x + 54*x^2; v[4] = 180*x^2 + 384*x^3; for(n = 4, N, my(i = n + 1); v[i] = (9*n^2 - 9)*x^2*v[i - 4] + (4*n^2 - 4)*x^3*deriv(v[i - 4], x) + (18*n^2 + 9*n - 9)*x^2*v[i - 3] + (8*n^2 + 4*n - 4)*x^3*deriv(v[i - 3], x) + ((4*n + 4)*x + (9*n^2 + 9*n)*x^2)*v[i - 2] - ((4*n + 4)*x^2 - (4*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (4*n + 4)*x*v[i - 1] - (4*n + 4)*x^2*deriv(v[i - 1], 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 T(n,k) read by rows; related to number of preorders.?
  - 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.