A144697 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,x + n\right)\,P_{n}(x) = \left(-2 - 3\,x + 2\,x^{2} - x^{3} - 2\,x^{4} - 2\,n - n x - 4\,n x^{2} - 2\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x - n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-x - x^{2} + x^{3} - x^{4} + 2\,x^{5} - n x - 2\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 6\,x + x^{2} + 2\,x^{3} + 3\,n + n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	11	28	9													48
3	26	159	148	27												360
4	57	708	1454	700	81											3000
5	120	2747	10716	10770	3124	243										27720
6	247	9796	66035	121280	70685	13468	729									282240
7	502	33087	361608	1109485	1151790	429513	56788	2187								3144960
8	1013	107788	1825680	8785604	14925730	9736692	2477432	235900	6561							38102400
9	2036	342603	8700928	62722196	163936632	172636954	75852512	13776612	969844	19683						498960000
10	4083	1070556	39764333	414838000	1593028542	2549745128	1791471514	556799472	74618015	3958108	59049					7025356800
11	8178	3305567	176169900	2590059855	14108312220	32782508094	34729733136	17146848150	3909739170	396423555	16068628	177147				105859353600
12	16369	10121140	762490226	15471554220	116305389375	378645883272	577895011260	427914475608	154287144255	26542976900	2076101234	64983100	531441			1699976678400
13	32752	30805371	3242180724	89289978278	906319599300	4019880962541	8528334175848	9052285381812	4877727820824	1323068341925	175582452132	10756440006	262058164	1594323		28986781824000
14	65519	93365908	13600052871	501520804768	6753062478659	39899267933652	114366877875579	167880742987488	129257165070909	52279892245372	10922880661829	1138305237888	55279026641	1054609948	4782969	523069747200000

Technical details

Node id: n-a5c58f4519622dba · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-5f3d6da2e469b205

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:    4,      3;
  2:   11,     28,       9;
  3:   26,    159,     148,       27;
  4:   57,    708,    1454,      700,        81;
  5:  120,   2747,   10716,    10770,      3124,       243;
  6:  247,   9796,   66035,   121280,     70685,     13468,      729;
  7:  502,  33087,  361608,  1109485,   1151790,    429513,    56788,     2187;
  8: 1013, 107788, 1825680,  8785604,  14925730,   9736692,  2477432,   235900,   6561;
  9: 2036, 342603, 8700928, 62722196, 163936632, 172636954, 75852512, 13776612, 969844, 19683;

%F Formula

The row polynomials satisfy (n+2*x)*P(n,x) = -(2*n+2+(2*n^2+n+3)*x+(n^3+n^2+4*n-2)*x^2+(3*n^2+2*n+1)*x^3+(2*n+2)*x^4)*P(n-2,x) - ((n+1)*x+(n^2+1)*x^2-(n^2-2*n+1)*x^3-(3*n-1)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n+1+(2*n^2+n+6)*x+(5*n+1)*x^2+2*x^3)*P(n-1,x) + (n*x-(n-2)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 + n + 3)*x + (n^3 + n^2 + 4*n - 2)*x^2 + (3*n^2 + 2*n + 1)*x^3 + (2*n + 2)*x^4)*P[n - 2, x] - ((n + 1)*x + (n^2 + 1)*x^2 - (n^2 - 2*n + 1)*x^3 - (3*n - 1)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n + 1 + (2*n^2 + n + 6)*x + (5*n + 1)*x^2 + 2*x^3)*P[n - 1, x] + (n*x - (n - 2)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 2 + (2*n^2 + n + 3)*x + (n^3 + n^2 + 4*n - 2)*x^2 + (3*n^2 + 2*n + 1)*x^3 + (2*n + 2)*x^4)*v[i - 2] - ((n + 1)*x + (n^2 + 1)*x^2 - (n^2 - 2*n + 1)*x^3 - (3*n - 1)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n + 1 + (2*n^2 + n + 6)*x + (5*n + 1)*x^2 + 2*x^3)*v[i - 1] + (n*x - (n - 2)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 2*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of 3-Eulerian numbers.?
  - 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.