A269945 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	1							1
3	1	2						3
4	1	10	3					14
5	1	42	42	4				89
6	1	170	441	120	5			737
7	1	682	4224	2508	275	6		7696
8	1	2730	39039	45760	10010	546	7	98093

All 15 rows

Technical details

Node id: n-3b3ead46f39c15a4 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-0e60f7e2ac8ceeab

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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: 0;
   2: 1;
   3: 1,     2;
   4: 1,    10,       3;
   5: 1,    42,      42,        4;
   6: 1,   170,     441,      120,       5;
   7: 1,   682,    4224,     2508,     275,       6;
   8: 1,  2730,   39039,    45760,   10010,     546,     7;
   9: 1, 10922,  355446,   784212,  307450,   31668,   980,    8;
  10: 1, 43690, 3215397, 13021320, 8666515, 1508988, 84966, 1632, 9;

%F Formula

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

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 2*x^2*D[P[n - 2, x], x] - x^3*D[P[n - 2, x], {x, 2}] + (1 + 2*x)*P[n - 1, x] + 3*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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -x^2*v[i - 2] - 2*x^2*deriv(v[i - 2], x) - x^3*deriv(deriv(v[i - 2], x), x) + (1 + 2*x)*v[i - 1] + 3*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) =…?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A036969 → θ+1

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.