A304334 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12\,x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} - 8\,x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{3}\right)\,P_{n-2}''(x) + \left(1 + 12\,x\right)\,P_{n-1}(x) + \left(3\,x + 4\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	1	12						13
3	1	60	180					241
4	1	252	2520	3360				6133
5	1	1020	26460	100800	75600			203881
6	1	4092	253440	2106720	4158000	1995840		8518093
7	1	16380	2342340	38438400	151351200	181621440	60540480	434310241

All 15 rows

Technical details

Node id: n-5af4ea1ce3f85b64 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-784ef30fa3492623

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 -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: 0;
  1: 1;
  2: 1,     12;
  3: 1,     60,       180;
  4: 1,    252,      2520,        3360;
  5: 1,   1020,     26460,      100800,        75600;
  6: 1,   4092,    253440,     2106720,      4158000,      1995840;
  7: 1,  16380,   2342340,    38438400,    151351200,    181621440,     60540480;
  8: 1,  65532,  21326760,   658738080,   4648644000,  10534043520,   8475667200,   2075673600;
  9: 1, 262140, 192923820, 10937908800, 131037706800, 501949768320, 734840346240, 423437414400, 79394515200;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -12*x^2*v[i - 2] - (4*x^2 + 8*x^3)*deriv(v[i - 2], x) - 2*x^3*deriv(deriv(v[i - 2], x), x) + (1 + 12*x)*v[i - 1] + (3*x + 4*x^2)*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 - 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.?
  - 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.