A299989 → D

nonnegative coefficients

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

Recurrence

\(\left(3 - x^{2} + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(9 + 24\,x + 22\,x^{2} + 8\,x^{3} + x^{4} + 12\,n x + 22\,n x^{2} + 12\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	3	8	3																											14
2	9	48	66	32	5																									160
3	27	216	513	544	285	72	7																							1664
4	81	864	2916	4800	4430	2400	756	128	9																					16384
5	243	3240	14175	31680	42150	35664	19670	7040	1575	200	11																			155648
6	729	11664	62694	177120	308475	356832	284452	158592	61695	16400	2838	288	13																	1441792
7	2187	40824	260253	889056	1915515	2816856	2950437	2251904	1264473	521640	156079	32928	4641	392	15															13107200
8	6561	139968	1032264	4136832	10625580	18978624	24678360	23999232	17720406	9999680	4308920	1405824	341068	59584	7080	512	17													117440512
9	19683	472392	3956283	18195840	54281340	114143904	177477804	210117888	192887082	138755440	78583626	35019648	12207468	3288096	670140	99840	10251	648	19											1040187392
10	59049	1574640	14762250	76632480	260417025	630275904	1141098840	1593872640	1752377490	1536189600	1082025692	614475840	281245770	103306560	30187800	6916608	1214565	157680	14250	800	21									9126805504
11	177147	5196312	53911737	311778720	1189017225	3255778152	6729167907	10838437632	13896686430	14389254000	12147106026	8408421696	4785223586	2238328400	857820150	267615744	67252119	13398264	2065965	237600	19173	968	23							79456894976
12	531441	17006112	193444524	1233258048	5215798170	15942285216	37045759212	67618260864	99114815943	118478859840	116738874648	95497923456	65147835532	37138081344	17687708280	7020969472	2311319439	626095008	137932476	24298560	3338874	344608	25116	1152	25					687194767360
13	1594323	55269864	683964567	4765490496	22136092290	74712573936	192836726082	393463204992	649256883561	880469698680	992663937837	938074400640	747231966156	503465318048	287396910060	138973985280	56819147957	19565993304	5640549057	1349325120	264522258	41753712	5173298	484224	32175	1352	27			5909874999296
14	4782969	178564176	2388295854	18054822240	91375670295	337539200544	958159385820	2161545699456	3966288006681	6018824251440	7645594678722	8202411296352	7479833258079	5823661480896	3882089966280	2218537707008	1086813379379	455689516464	163037372498	49537648160	12695023341	2717992992	479841180	68595072	7737275	662480	40446	1568	29	50577534877696

Technical details

Node id: n-a99de495fe940cd0 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-b1018962b1f2016f

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:     3,      8,       3;
  2:     9,     48,      66,       32,        5;
  3:    27,    216,     513,      544,      285,        72,         7;
  4:    81,    864,    2916,     4800,     4430,      2400,       756,       128,         9;
  5:   243,   3240,   14175,    31680,    42150,     35664,     19670,      7040,      1575,       200,       11;
  6:   729,  11664,   62694,   177120,   308475,    356832,    284452,    158592,     61695,     16400,     2838,      288,       13;
  7:  2187,  40824,  260253,   889056,  1915515,   2816856,   2950437,   2251904,   1264473,    521640,   156079,    32928,     4641,     392,     15;
  8:  6561, 139968, 1032264,  4136832, 10625580,  18978624,  24678360,  23999232,  17720406,   9999680,  4308920,  1405824,   341068,   59584,   7080,   512,    17;
  9: 19683, 472392, 3956283, 18195840, 54281340, 114143904, 177477804, 210117888, 192887082, 138755440, 78583626, 35019648, 12207468, 3288096, 670140, 99840, 10251, 648, 19;

%F Formula

The row polynomials satisfy (3+4*n*x+(2*n-1)*x^2)*P(n,x) = (9+(12*n+24)*x+(22*n+22)*x^2+(12*n+8)*x^3+(2*n+1)*x^4)*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[((9 + (12*n + 24)*x + (22*n + 22)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(3 + 4*n*x + (2*n - 1)*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] = ((9 + (12*n + 24)*x + (22*n + 22)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(3 + 4*n*x + (2*n - 1)*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 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…?
  - Parent-row provers of polytool polytool ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.