A071951 → θ+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	2							4
3	4	16	3						23
4	8	104	60	4					176
5	16	640	876	160	5				1697
6	32	3872	11472	4368	350	6			20100
7	64	23296	143472	102656	15960	672	7		286127
8	128	139904	1756608	2244416	607120	47376	1176	8	4796736

All 15 rows

Technical details

Node id: n-688020112ddc2e19 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-dbde9fb02b4eeb4a

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   1;
   2:   2,       2;
   3:   4,      16,         3;
   4:   8,     104,        60,         4;
   5:  16,     640,       876,       160,         5;
   6:  32,    3872,     11472,      4368,       350,         6;
   7:  64,   23296,    143472,    102656,     15960,       672,       7;
   8: 128,  139904,   1756608,   2244416,    607120,     47376,    1176,      8;
   9: 256,  839680,  21289152,  47230464,  21019120,   2718336,  121128,   1920,    9;
  10: 512, 5038592, 256729344, 972994816, 689611680, 139393056, 9954560, 276672, 2970, 10;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 3*x^2*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (2 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = -x^2*v[i - 2] - 3*x^2*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (2 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.?
  - 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.