A079510 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	9	15	6													30
3	44	112	92	24												272
4	265	900	1125	610	120											3020
5	1854	7926	13374	11106	4524	720										39504
6	14833	76615	163275	183505	114520	37548	5040									595336
7	133496	809824	2089520	2970080	2509240	1258496	346464	40320								10157440
8	1334961	9312912	28253106	48657420	51996105	35278236	14827932	3526416	362880							193549968
9	14684570	115911930	404653620	819701260	1061328450	910420770	517111280	187411320	39296160	3628800						4074148160
10	176214841	1553688103	6139447248	14313919466	21798515585	22649259171	16254020398	7951646428	2536564536	476166240	39916800					93889358816
11	2290792932	22325993328	98564410812	260132659128	455921577132	557034287712	483969739428	298922273112	128579021328	36667868832	6236542080	479001600				2351124167424
12	32071101049	342506959652	1671586013383	4928955676502	9778166727043	13746247314800	14038794987217	10492269831758	5693863570252	2187375072168	564466421376	87820986240	6227020800			63570351682240
13	481066515734	5588727857454	29889767280450	97429342637550	215924472957270	343523178693618	403564648608198	354395395641210	232590885750300	112650797456120	39126442662384	9226041318144	1323378120960	87178291200		1845801323790592
14	7697064251745	96665586508335	562388821664085	2008794774118275	4920914069975475	8744580985390965	11623093044057495	11737366636025745	9047586221902800	5296843827732600	2317936224180720	735071570822160	159652011692160	21251751494400	1307674368000	57281150264184960

Technical details

Node id: n-4770949c8ef98ba1 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-6ce805bb80e27667

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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;
  2:        9,        15,         6;
  3:       44,       112,        92,        24;
  4:      265,       900,      1125,       610,        120;
  5:     1854,      7926,     13374,     11106,       4524,       720;
  6:    14833,     76615,    163275,    183505,     114520,     37548,      5040;
  7:   133496,    809824,   2089520,   2970080,    2509240,   1258496,    346464,     40320;
  8:  1334961,   9312912,  28253106,  48657420,   51996105,  35278236,  14827932,   3526416,   362880;
  9: 14684570, 115911930, 404653620, 819701260, 1061328450, 910420770, 517111280, 187411320, 39296160, 3628800;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (n + 1)*x)*P[n - 2, x] + (n + 1 + (n + 1)*x)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (n + 1)*x)*v[i - 2] + (n + 1 + (n + 1)*x)*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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by 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; 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
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 (above)

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