A305540 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-8\,x + 22\,x^{2}\right)\,P_{n-4}(x) + \left(-2\,x + 26\,x^{3}\right)\,P_{n-4}'(x) + \left(-x^{2} + 4\,x^{3} + 5\,x^{4}\right)\,P_{n-4}''(x) + \left(1 + 8\,x\right)\,P_{n-2}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-2}'(x)\)

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

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

\(P_{2}(x) = 1 + 8\,x\)

\(P_{3}(x) = 1 + 16\,x + 27\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	8							9
3	1	16	27						44
4	1	24	54						79
5	1	40	189	192					422
6	1	56	324	384					765
7	1	88	837	2112	1500				4538
8	1	120	1350	3840	3000				8311
9	1	184	3105	14400	24000	12960			54650
10	1	248	4860	24960	45000	25920			100989
11	1	376	10557	79680	232500	285120	123480		731714
12	1	504	16254	134400	420000	544320	246960		1362439
13	1	760	34209	393792	1785000	3719520	3580920	1290240	10804442
14	1	1016	52164	653184	3150000	6894720	6914880	2580480	20246445

Technical details

Node id: n-54d7b058e1600620 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-afc8bef80990b82e

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,   4;
  2: 1,   8;
  3: 1,  16,   27;
  4: 1,  24,   54;
  5: 1,  40,  189,   192;
  6: 1,  56,  324,   384;
  7: 1,  88,  837,  2112,  1500;
  8: 1, 120, 1350,  3840,  3000;
  9: 1, 184, 3105, 14400, 24000, 12960;

%F Formula

The row polynomials satisfy P(n,x) = -(8*x-22*x^2)*P(n-4,x) - (2*x-26*x^3)*d/dx P(n-4,x) - (x^2-4*x^3-5*x^4)*d^2/dx^2 P(n-4,x) + (1+8*x)*P(n-2,x) + (2*x-4*x^2)*d/dx P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 8*x, P(3,x) = 1 + 16*x + 27*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 8*x;
P[3, x_] := 1 + 16*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[-(8*x - 22*x^2)*P[n - 4, x] - (2*x - 26*x^3)*D[P[n - 4, x], x] - (x^2 - 4*x^3 - 5*x^4)*D[P[n - 4, x], {x, 2}] + (1 + 8*x)*P[n - 2, x] + (2*x - 4*x^2)*D[P[n - 2, 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] = 1 + 4*x; v[3] = 1 + 8*x; v[4] = 1 + 16*x + 27*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(8*x - 22*x^2)*v[i - 4] - (2*x - 26*x^3)*deriv(v[i - 4], x) - (x^2 - 4*x^3 - 5*x^4)*deriv(deriv(v[i - 4], x), x) + (1 + 8*x)*v[i - 2] + (2*x - 4*x^2)*deriv(v[i - 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 Triangle read by rows: T(n,k) is the number of achiral loops (necklaces or bracelets) of length n using exactly k diffe…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.