A110552 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,n x - n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 12\,x - 4\,x^{3} - 2\,n x + 11\,n x^{2} + 2\,n x^{3} - 3\,n^{2} x^{2} + 3\,n^{2} x^{3} - n^{3} x^{3}\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{3} - 2\,x^{4} + 3\,n x^{2} + 3\,n x^{3} + n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	1	10	12													23
3	1	20	51	32												104
4	1	38	153	196	80											468
5	1	72	402	784	645	192										2096
6	1	138	990	2600	3225	1926	448									9328
7	1	268	2349	7760	12875	11556	5383	1024								41216
8	1	526	5439	21644	44975	53886	37681	14344	2304							180800
9	1	1040	12372	57568	143710	215376	200900	114752	36873	5120						787712
10	1	2066	27756	147792	430710	774900	903756	688416	331857	92170	11264					3410688
11	1	4116	61575	369120	1229850	2581992	3614142	3441600	2212245	921700	225291	24576				14686208
12	1	8214	135333	901780	3380850	8112852	13249698	15141456	12166605	6758950	2478201	540684	53248			62927872
13	1	16408	295110	2163568	9013675	24334992	45422916	60561600	58397031	40552600	19825366	6488208	1277965	114688		268464128
14	1	32794	639210	5112952	23432695	70295082	147615468	224933280	253046079	210869230	128863306	56230824	16613545	2981902	245760	1140912128

Technical details

Node id: n-35912eee1e284d52 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-0733988374790ceb

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: 1,    4;
  2: 1,   10,    12;
  3: 1,   20,    51,    32;
  4: 1,   38,   153,   196,     80;
  5: 1,   72,   402,   784,    645,    192;
  6: 1,  138,   990,  2600,   3225,   1926,    448;
  7: 1,  268,  2349,  7760,  12875,  11556,   5383,   1024;
  8: 1,  526,  5439, 21644,  44975,  53886,  37681,  14344,  2304;
  9: 1, 1040, 12372, 57568, 143710, 215376, 200900, 114752, 36873, 5120;

%F Formula

The row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = (2-(2*n-12)*x-(3*n^2-11*n)*x^2-(n^3-3*n^2-2*n+4)*x^3)*P(n-1,x) + (2*x+3*n*x^2+(n^2+3*n-4)*x^3+(n^2-2)*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[((2 - (2*n - 12)*x - (3*n^2 - 11*n)*x^2 - (n^3 - 3*n^2 - 2*n + 4)*x^3)*P[n - 1, x] + (2*x + 3*n*x^2 + (n^2 + 3*n - 4)*x^3 + (n^2 - 2)*x^4)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 - n)*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] = ((2 - (2*n - 12)*x - (3*n^2 - 11*n)*x^2 - (n^3 - 3*n^2 - 2*n + 4)*x^3)*v[i - 1] + (2*x + 3*n*x^2 + (n^2 + 3*n - 4)*x^3 + (n^2 - 2)*x^4)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 - n)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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.