A079510 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	4							6
2	9	30	24						63
3	44	224	368	192					828
4	265	1800	4500	4880	1920				13365
5	1854	15852	53496	88848	72384	23040			255474
6	14833	153230	653100	1468040	1832320	1201536	322560		5645619
7	133496	1619648	8358080	23760640	40147840	40271872	22173696	5160960	141626232

All 15 rows

Technical details

Node id: n-4d70fdb92495e233 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-e3ba5e0ca89c8da7

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,         4;
  2:        9,        30,         24;
  3:       44,       224,        368,        192;
  4:      265,      1800,       4500,       4880,        1920;
  5:     1854,     15852,      53496,      88848,       72384,       23040;
  6:    14833,    153230,     653100,    1468040,     1832320,     1201536,      322560;
  7:   133496,   1619648,    8358080,   23760640,    40147840,    40271872,    22173696,     5160960;
  8:  1334961,  18625824,  113012424,  389259360,   831937680,  1128903552,   948987648,   451381248,    92897280;
  9: 14684570, 231823860, 1618614480, 6557610080, 16981255200, 29133464640, 33095121920, 23988648960, 10059816960, 1857945600;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(2*n+2)*x)*P(n-2,x) + (n+1+(2*n+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (2*n + 2)*x)*P[n - 2, x] + (n + 1 + (2*n + 2)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (2*n + 2)*x)*v[i - 2] + (n + 1 + (2*n + 2)*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 (−∞, 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; 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