A200139 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	6	6	1													13
4	18	30	11	1												60
5	54	138	85	17	1											295
6	162	606	563	187	24	1										1543
7	486	2586	3421	1685	355	32	1									8566
8	1458	10830	19691	13531	4170	611	41	1								50333
9	4374	44778	109285	100877	42721	9058	980	51	1							312125
10	13122	183486	591203	714547	399924	115185	17878	1490	62	1						2036898
11	39366	747066	3139501	4878485	3514015	1321404	276087	32778	2172	74	1					13950949
12	118098	3027630	16444571	32410411	29476590	14085247	3806187	603867	56670	3060	87	1				100032419
13	354294	12228618	85250485	210907037	238746541	142158566	48340930	9844857	1227237	93390	4191	101	1			749156248
14	1062882	49268766	438481043	1350692707	1882132824	1376015069	577226936	146789500	23344464	2347917	147873	5605	116	1		5847515703
15	3188646	198137946	2241673981	8542637285	14525622475	12890253376	6571057493	2045121936	403578604	51519468	4270266	226343	7345	132	1	47477295297

Technical details

Node id: n-38c18259f535bbbd · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-e364eb52ea30b8b4

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     2,      1;
   3:     6,      6,      1;
   4:    18,     30,     11,      1;
   5:    54,    138,     85,     17,      1;
   6:   162,    606,    563,    187,     24,      1;
   7:   486,   2586,   3421,   1685,    355,     32,     1;
   8:  1458,  10830,  19691,  13531,   4170,    611,    41,    1;
   9:  4374,  44778, 109285, 100877,  42721,   9058,   980,   51,  1;
  10: 13122, 183486, 591203, 714547, 399924, 115185, 17878, 1490, 62, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = ((3*n - 6 + (n - 2)*x)*v[i - 1] + (n - 2)*x*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Ray basis map theorem

Proof chains ? JSON

Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
- from Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operat…?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.