A290315 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + x + n\right)\,P_{n}(x) = \left(6 - 111\,x + 1294\,x^{2} - 796\,x^{3} + 88\,x^{4} - 3\,n + 86\,n x - 1436\,n x^{2} + 552\,n x^{3} - 32\,n x^{4} - 16\,n^{2} x + 528\,n^{2} x^{2} - 96\,n^{2} x^{3} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x - 106\,x^{2} + 260\,x^{3} - 136\,x^{4} + 16\,x^{5} - 2\,n x + 84\,n x^{2} - 184\,n x^{3} + 48\,n x^{4} - 16\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-11 + 118\,x - 54\,x^{2} + 4\,x^{3} + 4\,n - 86\,n x + 20\,n x^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 14\,x^{2} - 4\,x^{3} + 2\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	2							5
3	17	40	12						69
4	67	634	644	120					1465
5	225	5968	20728	12672	1680				41273
6	707	42866	370088	700592	297264	30240			1441757
7	2161	266616	4725444	20749696	25456176	8121984	665280		59987357
8	6531	1527530	49573740	421961640	1145380880	1004138208	253638720	17297280	2893524529

All 15 rows

Technical details

Node id: n-5f1a3b5b4d2ae04f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ac32c7b310f5a466

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:     3,        2;
   3:    17,       40,         12;
   4:    67,      634,        644,         120;
   5:   225,     5968,      20728,       12672,         1680;
   6:   707,    42866,     370088,      700592,       297264,         30240;
   7:  2161,   266616,    4725444,    20749696,     25456176,       8121984,        665280;
   8:  6531,  1527530,   49573740,   421961640,   1145380880,    1004138208,     253638720,      17297280;
   9: 19649,  8329888,  459922800,  6797016320,  34443568480,   64500460032,   43022294784,    8917770240,    518918400;
  10: 59011, 44046562, 3935358416, 93925571680, 799078884640, 2718039420608, 3766872919296, 1996669859328, 348734718720, 17643225600;

%F Formula

The row polynomials satisfy (n-3+x)*P(n,x) = -(3*n-6+(16*n^2-86*n+111)*x+(64*n^3-528*n^2+1436*n-1294)*x^2+(96*n^2-552*n+796)*x^3+(32*n-88)*x^4)*P(n-2,x) - ((2*n-4)*x+(16*n^2-84*n+106)*x^2-(32*n^2-184*n+260)*x^3-(48*n-136)*x^4-16*x^5)*d/dx P(n-2,x) + (4*n-11+(16*n^2-86*n+118)*x+(20*n-54)*x^2+4*x^3)*P(n-1,x) + ((2*n-6)*x-(4*n-14)*x^2-4*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 + (16*n^2 - 86*n + 111)*x + (64*n^3 - 528*n^2 + 1436*n - 1294)*x^2 + (96*n^2 - 552*n + 796)*x^3 + (32*n - 88)*x^4)*P[n - 2, x] - ((2*n - 4)*x + (16*n^2 - 84*n + 106)*x^2 - (32*n^2 - 184*n + 260)*x^3 - (48*n - 136)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (4*n - 11 + (16*n^2 - 86*n + 118)*x + (20*n - 54)*x^2 + 4*x^3)*P[n - 1, x] + ((2*n - 6)*x - (4*n - 14)*x^2 - 4*x^3)*D[P[n - 1, x], x])/(n - 3 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(3*n - 6 + (16*n^2 - 86*n + 111)*x + (64*n^3 - 528*n^2 + 1436*n - 1294)*x^2 + (96*n^2 - 552*n + 796)*x^3 + (32*n - 88)*x^4)*v[i - 2] - ((2*n - 4)*x + (16*n^2 - 84*n + 106)*x^2 - (32*n^2 - 184*n + 260)*x^3 - (48*n - 136)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (4*n - 11 + (16*n^2 - 86*n + 118)*x + (20*n - 54)*x^2 + 4*x^3)*v[i - 1] + ((2*n - 6)*x - (4*n - 14)*x^2 - 4*x^3)*deriv(v[i - 1], x))/(n - 3 + 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.