A146987 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(36 + 108\,x + 81\,x^{2} - 12\,n - 36\,n x - 27\,n x^{2}\right)\,P_{n-2}(x) + \left(-24 - 36\,x + 8\,n + 12\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 3\,x\)

\(P_{2}(x) = 7 + 21\,x + 9\,x^{2}\)

\(P_{3}(x) = 26 + 117\,x + 135\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	3							5
2	7	21	9						37
3	26	117	135	27					305
4	124	744	1431	945	81				3325
5	680	5100	13842	16389	7371	243			43625
6	3952	35568	124632	214272	182007	61965	729		623125
7	23456	246288	1055808	2377080	2975130	1969029	541647	2187	9190625

All 15 rows

Technical details

Node id: n-c22b94fee16d0774 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-e3388fed2251ce3b

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,        3;
  2:      7,       21,        9;
  3:     26,      117,      135,        27;
  4:    124,      744,     1431,       945,        81;
  5:    680,     5100,    13842,     16389,      7371,       243;
  6:   3952,    35568,   124632,    214272,    182007,     61965,       729;
  7:  23456,   246288,  1055808,   2377080,   2975130,   1969029,    541647,      2187;
  8: 140224,  1682688,  8519184,  23667984,  39063060,  38372616,  20807847,   4817961,     6561;
  9: 840320, 11344320, 66176352, 218390256, 446737032, 580823460, 469220850, 215548533, 43164819, 19683;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(12*n-36+(36*n-108)*x+(27*n-81)*x^2)*P(n-2,x) + (8*n-24+(12*n-36)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 7 + 21*x + 9*x^2, P(3,x) = 26 + 117*x + 135*x^2 + 27*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 7 + 21*x + 9*x^2;
P[3, x_] := 26 + 117*x + 135*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 36 + (36*n - 108)*x + (27*n - 81)*x^2)*P[n - 2, x] + (8*n - 24 + (12*n - 36)*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] = 1; v[2] = 2 + 3*x; v[3] = 7 + 21*x + 9*x^2; v[4] = 26 + 117*x + 135*x^2 + 27*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(12*n - 36 + (36*n - 108)*x + (27*n - 81)*x^2)*v[i - 2] + (8*n - 24 + (12*n - 36)*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 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.?
  - 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