A249253 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 3 + 4\,x\)

\(P_{2}(x) = 12 + 20\,x\)

\(P_{3}(x) = 60 + 112\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	1					1
1	3	4				7
2	12	20				32
3	60	112	9			181
4	360	720	99			1179
5	2520	5280	945	16		8761
6	20160	43680	9180	304		73324
7	181440	403200	94500	4416	25	683581

All 15 rows

Technical details

Node id: n-6c1e32d966bf523c · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7e06e53fe8c962b0

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:        3,        4;
  2:       12,       20;
  3:       60,      112,        9;
  4:      360,      720,       99;
  5:     2520,     5280,      945,     16;
  6:    20160,    43680,     9180,    304;
  7:   181440,   403200,    94500,   4416,    25;
  8:  1814400,  4112640,  1043280,  60480,   725;
  9: 19958400, 45964800, 12383280, 833280, 14875, 36;

%F Formula

The row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - 4*n*x*P(n-3,x) + 4*n*x^2*d/dx P(n-3,x) - (n^2+n-4*x)*P(n-2,x) - 4*x^2*d/dx P(n-2,x) + (2*n+2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 12 + 20*x, P(3,x) = 60 + 112*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 12 + 20*x;
P[3, x_] := 60 + 112*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - 4*n*x*P[n - 3, x] + 4*n*x^2*D[P[n - 3, x], x] - (n^2 + n - 4*x)*P[n - 2, x] - 4*x^2*D[P[n - 2, x], x] + (2*n + 2)*P[n - 1, 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] = 3 + 4*x; v[3] = 12 + 20*x; v[4] = 60 + 112*x + 9*x^2; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - 4*n*x*v[i - 3] + 4*n*x^2*deriv(v[i - 3], x) - (n^2 + n - 4*x)*v[i - 2] - 4*x^2*deriv(v[i - 2], x) + (2*n + 2)*v[i - 1]); 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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - 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.