A079510 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	8							10
2	9	60	96						165
3	44	448	1472	1536					3500
4	265	3600	18000	39040	30720				91625
5	1854	31704	213984	710784	1158144	737280			2853750
6	14833	306460	2612400	11744320	29317120	38449152	20643840		103088125
7	133496	3239296	33432320	190085120	642365440	1288699904	1419116544	660602880	4237675000

All 15 rows

Technical details

Node id: n-f209d0e79422f087 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-28fe48d946165a82

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,         8;
  2:        9,        60,         96;
  3:       44,       448,       1472,        1536;
  4:      265,      3600,      18000,       39040,        30720;
  5:     1854,     31704,     213984,      710784,      1158144,       737280;
  6:    14833,    306460,    2612400,    11744320,     29317120,     38449152,      20643840;
  7:   133496,   3239296,   33432320,   190085120,    642365440,   1288699904,    1419116544,     660602880;
  8:  1334961,  37251648,  452049696,  3114074880,  13311002880,  36124913664,   60735209472,   57776799744,   23781703680;
  9: 14684570, 463647720, 6474457920, 52460880640, 271700083200, 932270868480, 2118087802880, 3070547066880, 2575313141760, 951268147200;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(4*n+4)*x)*P(n-2,x) + (n+1+(4*n+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[(n + 1 + (4*n + 4)*x)*P[n - 2, x] + (n + 1 + (4*n + 4)*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (4*n + 4)*x)*v[i - 2] + (n + 1 + (4*n + 4)*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