A153518 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-15 - 60\,x + 5\,n + 20\,n x\right)\,P_{n-2}(x) + \left(-6 - 12\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 10 + 20\,x\)

\(P_{2}(x) = 50 + 200\,x + 32\,x^{2}\)

\(P_{3}(x) = 250 + 1500\,x + 2064\,x^{2} + 128\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	10	20							30
2	50	200	32						282
3	250	1500	2064	128					3942
4	750	6000	14288	9152	512				30702
5	2750	27500	92896	117376	40192	2048			282762
6	9250	111000	487232	937856	735488	175104	8192		2464122
7	32250	451500	2432944	6269440	7770880	4106240	757760	32768	21853782

All 15 rows

Technical details

Node id: n-98886efaea10cd66 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-2aa284a691d23675

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,      20;
  2:     50,     200,       32;
  3:    250,    1500,     2064,       128;
  4:    750,    6000,    14288,      9152,       512;
  5:   2750,   27500,    92896,    117376,     40192,      2048;
  6:   9250,  111000,   487232,    937856,    735488,    175104,      8192;
  7:  32250,  451500,  2432944,   6269440,   7770880,   4106240,    757760,     32768;
  8: 110750, 1772000, 11328048,  36704576,  63054080,  54881280,  21483520,   3260416,   131072;
  9: 382750, 6889500, 50938816, 198727424, 437169664, 537927680, 348405760, 107773952, 13959168, 524288;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(20*n-60)*x)*P(n-2,x) + (2*n-6+(4*n-12)*x)*P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 10 + 20*x, P(2,x) = 50 + 200*x + 32*x^2, P(3,x) = 250 + 1500*x + 2064*x^2 + 128*x^3.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 20*x;
P[2, x_] := 50 + 200*x + 32*x^2;
P[3, x_] := 250 + 1500*x + 2064*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (20*n - 60)*x)*P[n - 2, x] + (2*n - 6 + (4*n - 12)*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 + 20*x; v[3] = 50 + 200*x + 32*x^2; v[4] = 250 + 1500*x + 2064*x^2 + 128*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (20*n - 60)*x)*v[i - 2] + (2*n - 6 + (4*n - 12)*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