A166962 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 3\,n\right)\,P_{n}(x) = \left(16 - 32\,x + 80\,x^{2} - 24\,n + 60\,n x - 198\,n x^{2} - 18\,n^{2} x + 135\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(26\,x - 106\,x^{2} + 30\,x^{3} - 39\,n x + 219\,n x^{2} - 63\,n x^{3} - 90\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(6\,x^{2} - 30\,x^{3} - 9\,n x^{2} + 63\,n x^{3} - 27\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-42 + 50\,x + 27\,n - 60\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-65\,x + 15\,x^{2} + 39\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x) + \left(-15\,x^{2} + 9\,n x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	12	2						14
3	103	138	3					244
4	834	4340	948	4				6126
5	6685	107820	83490	5316	5			203316
6	53496	2438030	4956900	1124780	27060	6		8600272
7	427987	52910502	245817585	139202700	12439585	130686	7	450929052

All 15 rows

Technical details

Node id: n-7cb278ca98b33a0d · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ff3c128a3f309512

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:       12,           2;
  3:      103,         138,            3;
  4:      834,        4340,          948,             4;
  5:     6685,      107820,        83490,          5316,            5;
  6:    53496,     2438030,      4956900,       1124780,        27060,           6;
  7:   427987,    52910502,    245817585,     139202700,     12439585,      130686,          7;
  8:  3423918,  1127384048,  11102555448,   13309013640,   2934233910,   121873752,     611100,       8;
  9: 27391369, 23825717400, 476232663288, 1101940402824, 497957927010, 51420015936, 1102855908, 2795400, 9;

%F Formula

The row polynomials satisfy (3*n-5)*P(n,x) = -(24*n-16+(18*n^2-60*n+32)*x+(27*n^3-135*n^2+198*n-80)*x^2)*P(n-2,x) - ((39*n-26)*x+(90*n^2-219*n+106)*x^2-(27*n^2-63*n+30)*x^3)*d/dx P(n-2,x) - ((9*n-6)*x^2+(27*n^2-63*n+30)*x^3)*d^2/dx^2 P(n-2,x) + (27*n-42+(18*n^2-60*n+50)*x)*P(n-1,x) + ((39*n-65)*x-(9*n-15)*x^2)*d/dx P(n-1,x) + (9*n-15)*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[Together[(-(24*n - 16 + (18*n^2 - 60*n + 32)*x + (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*P[n - 2, x] - ((39*n - 26)*x + (90*n^2 - 219*n + 106)*x^2 - (27*n^2 - 63*n + 30)*x^3)*D[P[n - 2, x], x] - ((9*n - 6)*x^2 + (27*n^2 - 63*n + 30)*x^3)*D[P[n - 2, x], {x, 2}] + (27*n - 42 + (18*n^2 - 60*n + 50)*x)*P[n - 1, x] + ((39*n - 65)*x - (9*n - 15)*x^2)*D[P[n - 1, x], x] + (9*n - 15)*x^2*D[P[n - 1, x], {x, 2}])/(3*n - 5)]];
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] = (-(24*n - 16 + (18*n^2 - 60*n + 32)*x + (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*v[i - 2] - ((39*n - 26)*x + (90*n^2 - 219*n + 106)*x^2 - (27*n^2 - 63*n + 30)*x^3)*deriv(v[i - 2], x) - ((9*n - 6)*x^2 + (27*n^2 - 63*n + 30)*x^3)*deriv(deriv(v[i - 2], x), x) + (27*n - 42 + (18*n^2 - 60*n + 50)*x)*v[i - 1] + ((39*n - 65)*x - (9*n - 15)*x^2)*deriv(v[i - 1], x) + (9*n - 15)*x^2*deriv(deriv(v[i - 1], x), x))/(3*n - 5)); 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 Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=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.