A008306 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	9	6							15
3	44	40							84
4	265	320	60						645
5	1854	2688	840						5382
6	14833	24766	10780	840					51219
7	133496	249296	135968	20160					538920
8	1334961	2733552	1766520	383040	15120				6233193
9	14684570	32498400	24010800	6751360	554400				78499530
10	176214841	416920614	343688664	117341840	14691600	332640			1069190199
11	2290792932	5745457848	5192355168	2065377600	344984640	17297280			15656265468
12	32071101049	84692505872	82808505780	37311257984	7726678960	611170560	8648640		245229868845
13	481066515734	1330273694176	1392884873016	696658842880	170833863200	18458119680	605404800		4090781313486
14	7697064251745	22186626032190	24676175858100	13493806686360	3797745871920	517839722400	27545918400	259459200	72397063800315

Technical details

Node id: n-850ad8a1478de773 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-96601b99a53d74c6

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,        6;
  3:       44,       40;
  4:      265,      320,       60;
  5:     1854,     2688,      840;
  6:    14833,    24766,    10780,     840;
  7:   133496,   249296,   135968,   20160;
  8:  1334961,  2733552,  1766520,  383040,  15120;
  9: 14684570, 32498400, 24010800, 6751360, 554400;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(2*n+2)*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 + (2*n + 2)*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 + (2*n + 2)*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