A249253 → basis A008277

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	1							5
2	17	5							22
3	89	31	1						121
4	551	213	11						775
5	3946	1642	111	1					5700
6	32119	14113	1134	19					47385
7	293017	134247	12181	286	1				439732
8	2962289	1402815	139325	4070	29				4508528
9	32878196	15992476	1703365	58095	610	1			50632743
10	397500641	197677631	22261170	852745	11535	41			618303763
11	5200386529	2634672351	310497781	13021430	211100	1149	1		8158790341
12	73202912047	37678268817	4611430075	207972170	3865820	27867	55		115704476851
13	1103244067234	575643763486	72737616819	3482166051	72064230	635999	1981	1	1755180315801
14	17725107987791	9358859665457	1215314428146	61157975571	1380328950	14209006	59948	71	28361834654940

Technical details

Node id: n-acced3d80d92e3dd · revision 0 · JSON
Definition source: derived:basis_map_A008277
Fingerprint: f6-bf121047df3ade02

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        4,        1;
  2:       17,        5;
  3:       89,       31,       1;
  4:      551,      213,      11;
  5:     3946,     1642,     111,     1;
  6:    32119,    14113,    1134,    19;
  7:   293017,   134247,   12181,   286,   1;
  8:  2962289,  1402815,  139325,  4070,  29;
  9: 32878196, 15992476, 1703365, 58095, 610, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + x*d/dx P(n-2,x) + (n+2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (n + 2)*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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (n + 2)*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 Ray basis map theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Ray basis map theorem
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - 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
[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