A131222 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-288 + 408\,n - 212\,n^{2} + 48\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(-180 + 24\,x + 222\,n - 20\,n x - 90\,n^{2} + 4\,n^{2} x + 12\,n^{3}\right)\,P_{n-3}(x) + \left(-52 + 12\,x - x^{2} + 52\,n - 6\,n x - 13\,n^{2}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 3 + 5\,x + x^{2}\)

\(P_{3}(x) = 14 + 32\,x + 12\,x^{2} + x^{3}\)

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	1														2
2	3	5	1													9
3	14	32	12	1												59
4	90	256	137	22	1											506
5	744	2484	1695	395	35	1										5354
6	7560	28508	23299	7015	910	51	1									67344
7	91440	379224	356664	130144	21980	1813	70	1								981336
8	1285200	5751912	6055944	2571352	530089	57204	3262	92	1							16255056
9	20603520	98143200	113381556	54466460	13156017	1741089	130326	5442	117	1						301627728
10	371226240	1862798112	2326159116	1238571860	340745075	53079453	4888443	268590	8565	145	1					6197745600
11	7428153600	38960619840	51984677448	30206143176	9270786250	1652292620	179693976	12176373	511830	12870	176	1				139695068160
12	163459296000	890741177280	1258466203224	788374407832	265711505026	53066453880	6624203333	531314058	27593313	915860	18623	210	1			3427003088640
13	3923502105600	22107997336320	32835210587712	21963798174240	8029025848844	1768535928016	248170084019	22901616023	1410380829	57918003	1556269	26117	247	1		90900611562240
14	102017281766400	592147653016320	919132002466560	651364845130272	255706186006908	61348524741504	9524618534431	990415882027	70303461800	3428353929	114101988	2532621	35672	287	1	2592305376030720

Technical details

Node id: n-226363e48ade0e33 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-b371c576b456bdba

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,        1;
  2:        3,        5,         1;
  3:       14,       32,        12,        1;
  4:       90,      256,       137,       22,        1;
  5:      744,     2484,      1695,      395,       35,       1;
  6:     7560,    28508,     23299,     7015,      910,      51,      1;
  7:    91440,   379224,    356664,   130144,    21980,    1813,     70,    1;
  8:  1285200,  5751912,   6055944,  2571352,   530089,   57204,   3262,   92,   1;
  9: 20603520, 98143200, 113381556, 54466460, 13156017, 1741089, 130326, 5442, 117, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^4-48*n^3+212*n^2-408*n+288)*P(n-4,x) + (12*n^3-90*n^2+222*n-180+(4*n^2-20*n+24)*x)*P(n-3,x) - (13*n^2-52*n+52+(6*n-12)*x+x^2)*P(n-2,x) + (6*n-9+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 3 + 5*x + x^2, P(3,x) = 14 + 32*x + 12*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 3 + 5*x + x^2;
P[3, x_] := 14 + 32*x + 12*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*P[n - 4, x] + (12*n^3 - 90*n^2 + 222*n - 180 + (4*n^2 - 20*n + 24)*x)*P[n - 3, x] - (13*n^2 - 52*n + 52 + (6*n - 12)*x + x^2)*P[n - 2, x] + (6*n - 9 + 2*x)*P[n - 1, 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] = 1 + x; v[3] = 3 + 5*x + x^2; v[4] = 14 + 32*x + 12*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 48*n^3 + 212*n^2 - 408*n + 288)*v[i - 4] + (12*n^3 - 90*n^2 + 222*n - 180 + (4*n^2 - 20*n + 24)*x)*v[i - 3] - (13*n^2 - 52*n + 52 + (6*n - 12)*x + x^2)*v[i - 2] + (6*n - 9 + 2*x)*v[i - 1]); 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 Exponential Riordan array [1, log((1-x)/(1-2x))].?
  - 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.