A008306 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	9	9							18
3	44	60							104
4	265	480	135						880
5	1854	4032	1890						7776
6	14833	37149	24255	2835					79072
7	133496	373944	305928	68040					881408
8	1334961	4100328	3974670	1292760	76545				10779264
9	14684570	48747600	54024300	22785840	2806650				143048960
10	176214841	625380921	773299494	396028710	74376225	2525985			2047826176
11	2290792932	8618186772	11682799128	6970649400	1746484740	131351220			31440264192
12	32071101049	127038758808	186319138005	125925495696	39116312235	4641076440	98513415		515210395648
13	481066515734	1995410541264	3133990964286	2351223594720	864846432450	140166346320	6895939050		8973600333824
14	7697064251745	33279939048285	55521395680725	45541597566465	19226088476595	3932345391975	313765226775	4433103675	165516628746240

Technical details

Node id: n-0326348c264dd39b · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-4d88b1184575328c

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:        2;
  2:        9,        9;
  3:       44,       60;
  4:      265,      480,      135;
  5:     1854,     4032,     1890;
  6:    14833,    37149,    24255,     2835;
  7:   133496,   373944,   305928,    68040;
  8:  1334961,  4100328,  3974670,  1292760,   76545;
  9: 14684570, 48747600, 54024300, 22785840, 2806650;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (3*n + 3)*x)*P[n - 2, x] + (n + 1)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (3*n + 3)*x)*v[i - 2] + (n + 1)*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 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: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows; related to number of preorders.?
    
    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