A136394 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x + 4\,n x\right)\,P_{n-2}(x) + \left(n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	4							6
3	6	20							26
4	24	104	48						176
5	120	616	560						1296
6	720	4176	5440	960					11296
7	5040	32112	52864	20160					110176
8	40320	277056	539840	313600	26880				1197696
9	362880	2654784	5886144	4514048	887040				14304896
10	3628800	27999360	68835456	64574720	20160000	967680			186166016
11	39916800	322508160	863381376	945767680	402321920	46126080			2620022016
12	479001600	4029765120	11592548352	14377972224	7669150720	1440552960	42577920		39631568896
13	6227020800	54302952960	166183520256	228355944960	145095808000	38038640640	2767564800		640971452416
14	87178291200	785149424640	2536117069824	3799795743744	2778995867648	931336806400	113654661120	2214051840	11034441916416

Technical details

Node id: n-2e619f362341fd8a · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-045f1532ff1709f8

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,       4;
  3:      6,      20;
  4:     24,     104,      48;
  5:    120,     616,     560;
  6:    720,    4176,    5440,     960;
  7:   5040,   32112,   52864,   20160;
  8:  40320,  277056,  539840,  313600,  26880;
  9: 362880, 2654784, 5886144, 4514048, 887040;

%F Formula

The row polynomials satisfy P(n,x) = (4*n-4)*x*P(n-2,x) + n*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[(4*n - 4)*x*P[n - 2, x] + n*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (4*n - 4)*x*v[i - 2] + n*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 read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=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

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