A090452 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(2 - 2\,x^{2} + 4\,n + 9\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 4\,x^{2} - 2\,x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	2	3	1																											6
2	5	15	16	7	1																									44
3	14	63	114	105	51	12	1																							360
4	42	252	644	910	771	396	118	18	1																					3152
5	132	990	3270	6240	7580	6083	3235	1110	230	25	1																			28896
6	429	3861	15642	37620	59625	65331	50464	27483	10365	2600	402	33	1																	273856
7	1430	15015	72072	209209	409332	569415	578690	434493	240751	97188	27825	5390	651	42	1															2661504
8	4862	58344	323752	1101100	2564562	4328324	5462072	5241060	3849715	2161288	918484	289884	65828	10220	996	52	1													26380544
9	16796	226746	1428102	5569200	15049008	29875482	45072846	52723476	48344922	34895685	19793907	8757108	2979312	762048	141552	18096	1458	63	1											265655808
10	58786	881790	6214520	27335490	84075030	191962368	337119120	465527790	512370495	452841675	322239204	184309005	84223665	30417390	8536920	1816248	282375	30345	2060	75	1									2710244352
11	208012	3432198	26761196	131070170	452119250	1167203136	2339265654	3725785008	4788113220	5012613375	4297304088	3021922137	1740293049	816717990	309674475	93647466	22169994	4001613	530090	48675	2827	88	1							27952883712
12	742900	13372200	114287736	616815012	2358346386	6792767520	15300852380	27618023268	40604633586	49159626648	49352002920	41238216630	28710585099	16629143688	7979929254	3150785550	1013169465	261545328	53136116	8264718	946275	75240	3786	102	1					290977271808
13	2674440	52151580	484816540	2858679200	11999499096	38148880770	95416220042	192514458708	318754907900	438294903618	504554375754	488794023744	399589566168	275827551795	160539021279	78511523610	32078801898	10860320757	3012028305	673878348	119118428	16168867	1619007	112710	4966	117	1			3053105307648
14	9694845	203591745	2045277990	13081354650	59805790135	208020580495	572056089332	1275982857006	2349839462971	3618158641097	4699678925942	5181437896908	4867860327057	3904937056923	2675584398912	1563800707875	777335641173	327060707064	115687505934	34086125073	8263934679	1622156250	252317156	30204083	2671123	164346	6398	133	1	32256844087296

Technical details

Node id: n-fb5f5aabac06a450 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-2df49debe483a37b

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:     2,      3,       1;
  2:     5,     15,      16,       7,        1;
  3:    14,     63,     114,     105,       51,       12,        1;
  4:    42,    252,     644,     910,      771,      396,      118,       18,        1;
  5:   132,    990,    3270,    6240,     7580,     6083,     3235,     1110,      230,       25,        1;
  6:   429,   3861,   15642,   37620,    59625,    65331,    50464,    27483,    10365,     2600,      402,      33,       1;
  7:  1430,  15015,   72072,  209209,   409332,   569415,   578690,   434493,   240751,    97188,    27825,    5390,     651,     42,      1;
  8:  4862,  58344,  323752, 1101100,  2564562,  4328324,  5462072,  5241060,  3849715,  2161288,   918484,  289884,   65828,  10220,    996,    52,    1;
  9: 16796, 226746, 1428102, 5569200, 15049008, 29875482, 45072846, 52723476, 48344922, 34895685, 19793907, 8757108, 2979312, 762048, 141552, 18096, 1458, 63, 1;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (4*n+2+9*n*x+(5*n-2)*x^2)*P(n-1,x) - (2*x+4*x^2+2*x^3)*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[((4*n + 2 + 9*n*x + (5*n - 2)*x^2)*P[n - 1, x] - (2*x + 4*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 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] = ((4*n + 2 + 9*n*x + (5*n - 2)*x^2)*v[i - 1] - (2*x + 4*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Scaled array A078740 ((3,2)-Stirling2).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A090452: Scaled array A078740 ((3,2)-Stirling2).

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