A079510 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	6							8
2	9	45	54						108
3	44	336	828	648					1856
4	265	2700	10125	16470	9720				39280
5	1854	23778	120366	299862	366444	174960			987264
6	14833	229845	1469475	4954635	9276120	9124164	3674160		28743232
7	133496	2429472	18805680	80192160	203248440	305814528	252572256	88179840	951375872

All 15 rows

Technical details

Node id: n-9e70cba966d7e1d5 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-022c58b01aaed433

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,         6;
  2:        9,        45,         54;
  3:       44,       336,        828,         648;
  4:      265,      2700,      10125,       16470,        9720;
  5:     1854,     23778,     120366,      299862,      366444,       174960;
  6:    14833,    229845,    1469475,     4954635,     9276120,      9124164,      3674160;
  7:   133496,   2429472,   18805680,    80192160,   203248440,    305814528,    252572256,     88179840;
  8:  1334961,  27938736,  254277954,  1313750340,  4211684505,   8572611348,  10809562428,   7712271792,   2380855680;
  9: 14684570, 347735790, 3641882580, 22131934020, 85967604450, 221232247110, 376974123120, 409868556840, 257822105760, 71425670400;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(3*n+3)*x)*P(n-2,x) + (n+1+(3*n+3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (3*n + 3)*x)*P[n - 2, x] + (n + 1 + (3*n + 3)*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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (3*n + 3)*x)*v[i - 2] + (n + 1 + (3*n + 3)*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