A049403 → reverse

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	P_n(1)
0	1	3					4
1	1	6	3				10
2	1	10	15				26
3	1	15	45	15			76
4	1	21	105	105			232
5	1	28	210	420	105		764
6	1	36	378	1260	945		2620
7	1	45	630	3150	4725	945	9496

All 15 rows

Technical details

Node id: n-6844deeae511b843 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-eccca8d515312e3b

Observed on the stored rows ?

Degrees: 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8 (half affine: slope 1, intercept 3)
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,  3;
  1: 1,  6,    3;
  2: 1, 10,   15;
  3: 1, 15,   45,    15;
  4: 1, 21,  105,   105;
  5: 1, 28,  210,   420,   105;
  6: 1, 36,  378,  1260,   945;
  7: 1, 45,  630,  3150,  4725,   945;
  8: 1, 55,  990,  6930, 17325, 10395;
  9: 1, 66, 1485, 13860, 51975, 62370, 10395;

%F Formula

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

%t Mathematica

P[0, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(1 + (n + 2)*x)*P[n - 1, x] - 2*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + (n + 2)*x)*v[i - 1] - 2*x^2*deriv(v[i - 1], x)); 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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