A110552 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x + x^{2} - n + 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-8 - 7\,x - 2\,x^{2} + 3\,x^{3} + 5\,n + 2\,n x + 7\,n x^{2} - n x^{3} + 4\,n^{2} x - 2\,n^{2} x^{2} - n^{3} x\right)\,P_{n-1}(x) + \left(-2 - 3\,x - x^{2} + x^{3} + x^{4} + n + n x + 2\,n x^{2} + 2\,n x^{3} + n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	6	13	4													23
3	11	44	41	8												104
4	20	121	198	113	16											468
5	37	304	722	712	289	32										2096
6	70	729	2280	3230	2250	705	64									9328
7	135	1700	6603	12240	12205	6540	1665	128								41216
8	264	3889	18046	41391	53900	41279	17934	3841	256							180800
9	521	8768	47300	129360	208222	208096	129108	47120	8705	512						787712
10	1034	19537	120096	381516	731892	903798	731472	381036	119826	19457	1024					3410688
11	2059	43108	297365	1076160	2397630	3528168	3527706	2396640	1075335	296980	43009	2048				14686208
12	4108	94329	721446	2930125	7436880	12709026	15141588	12707442	7434900	2928805	720918	94209	4096			62927872
13	8205	204944	1721046	7751832	22081895	42989760	59480388	59478672	42985899	22078320	7749830	1720344	204801	8192		268464128
14	16398	442537	4047784	20024394	63265774	138243391	217904544	253046508	217898538	138235383	63259768	20021482	4046874	442369	16384	1140912128

Technical details

Node id: n-68b03df3e3ed67f3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8207f97d812804d8

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 0)
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:   1;
  1:   3,    2;
  2:   6,   13,     4;
  3:  11,   44,    41,      8;
  4:  20,  121,   198,    113,     16;
  5:  37,  304,   722,    712,    289,     32;
  6:  70,  729,  2280,   3230,   2250,    705,     64;
  7: 135, 1700,  6603,  12240,  12205,   6540,   1665,   128;
  8: 264, 3889, 18046,  41391,  53900,  41279,  17934,  3841,  256;
  9: 521, 8768, 47300, 129360, 208222, 208096, 129108, 47120, 8705, 512;

%F Formula

The row polynomials satisfy (n^2-n-1+(2*n-2)*x+x^2)*P(n,x) = (5*n-8-(n^3-4*n^2-2*n+7)*x-(2*n^2-7*n+2)*x^2-(n-3)*x^3)*P(n-1,x) + (n-2+(n^2+n-3)*x+(n^2+2*n-1)*x^2+(2*n+1)*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 8 - (n^3 - 4*n^2 - 2*n + 7)*x - (2*n^2 - 7*n + 2)*x^2 - (n - 3)*x^3)*P[n - 1, x] + (n - 2 + (n^2 + n - 3)*x + (n^2 + 2*n - 1)*x^2 + (2*n + 1)*x^3 + x^4)*D[P[n - 1, x], x])/(n^2 - n - 1 + (2*n - 2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n - 8 - (n^3 - 4*n^2 - 2*n + 7)*x - (2*n^2 - 7*n + 2)*x^2 - (n - 3)*x^3)*v[i - 1] + (n - 2 + (n^2 + n - 3)*x + (n^2 + 2*n - 1)*x^2 + (2*n + 1)*x^3 + x^4)*deriv(v[i - 1], x))/(n^2 - n - 1 + (2*n - 2)*x + x^2)); 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 A triangular array related to A077028 and distributing the values of A007582.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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.