A185263 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	2							4
3	6	10							16
4	24	62	32						118
5	120	406	336						862
6	720	3096	4036	1440					9292
7	5040	25992	44004	24352					99388
8	40320	245232	530196	466528	129024				1411300
9	362880	2529648	6538356	7407136	3091968				19929988
10	3628800	28683360	87593976	127131592	84970176	19353600			351361504
11	39916800	351804960	1229418216	2125800872	1814611136	609822720			6171374704
12	479001600	4672149120	18527833152	37901480856	41555446336	22491021312	4379443200		130006375576
13	6227020800	66485957760	293711969088	686364988344	893500095872	613114863104	172779061248		2732183956216
14	87178291200	1013311434240	4955936003712	13150167273408	20275031605264	17910101536512	8170420368384	1394852659200	66956999171920

Technical details

Node id: n-20c6b6131d05ee73 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-0b8646f5e24f04bb

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:      2,       2;
  3:      6,      10;
  4:     24,      62,      32;
  5:    120,     406,     336;
  6:    720,    3096,    4036,    1440;
  7:   5040,   25992,   44004,   24352;
  8:  40320,  245232,  530196,  466528,  129024;
  9: 362880, 2529648, 6538356, 7407136, 3091968;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = (n^3-n^2-n+1+(2*n^3-2*n^2)*x)*P(n-2,x) + (2*n+1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n^2 - n + 1 + (2*n^3 - 2*n^2)*x)*P[n - 2, x] + (2*n + 1)*P[n - 1, 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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - n^2 - n + 1 + (2*n^3 - 2*n^2)*x)*v[i - 2] + (2*n + 1)*v[i - 1])/(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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t)…?
  - 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