A153518 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-15 - 15\,x + 5\,n + 5\,n x\right)\,P_{n-2}(x) + \left(-6 - 3\,x + 2\,n + n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 10 + 5\,x\)

\(P_{2}(x) = 50 + 50\,x + 2\,x^{2}\)

\(P_{3}(x) = 250 + 375\,x + 129\,x^{2} + 2\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	10	5							15
2	50	50	2						102
3	250	375	129	2					756
4	750	1500	893	143	2				3288
5	2750	6875	5806	1834	157	2			17424
6	9250	27750	30452	14654	2873	171	2		85152
7	32250	112875	152059	97960	30355	4010	185	2	429696

All 15 rows

Technical details

Node id: n-66e9d524a577c885 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-98d6195e3aed088b

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:      2;
  1:     10,       5;
  2:     50,      50,       2;
  3:    250,     375,     129,       2;
  4:    750,    1500,     893,     143,       2;
  5:   2750,    6875,    5806,    1834,     157,      2;
  6:   9250,   27750,   30452,   14654,    2873,    171,     2;
  7:  32250,  112875,  152059,   97960,   30355,   4010,   185,    2;
  8: 110750,  443000,  708003,  573509,  246305,  53595,  5245,  199,   2;
  9: 382750, 1722375, 3183676, 3105116, 1707694, 525320, 85060, 6578, 213, 2;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-2,x) + (2*n-6+(n-3)*x)*P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 10 + 5*x, P(2,x) = 50 + 50*x + 2*x^2, P(3,x) = 250 + 375*x + 129*x^2 + 2*x^3.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 5*x;
P[2, x_] := 50 + 50*x + 2*x^2;
P[3, x_] := 250 + 375*x + 129*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 2, x] + (2*n - 6 + (n - 3)*x)*P[n - 1, x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 5*x; v[3] = 50 + 50*x + 2*x^2; v[4] = 250 + 375*x + 129*x^2 + 2*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 2] + (2*n - 6 + (n - 3)*x)*v[i - 1])/(n - 3)); 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 ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.?
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - 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