A049403 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

\(P_{0}(x) = 4 + 5\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	4	5	1								10
1	10	18	9	1							38
2	26	64	51	14	1						156
3	76	230	250	115	20	1					692
4	232	846	1170	755	225	27	1				3256
5	764	3220	5376	4480	1925	399	35	1			16200
6	2620	12608	24724	25256	14245	4340	658	44	1		84496
7	9496	51084	114696	138684	97146	39501	8904	1026	54	1	460592

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

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

\(P_{0}(x) = 4 + 5\,x + x^{2}\)

\(P_{1}(x) = 10 + 18\,x + 9\,x^{2} + x^{3}\)

Technical details

Node id: n-ea69de4dd6061116 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-55091369e340a36e)
Fingerprint: f6-52e3702c84a5e3fb

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:      4,      5,       1;
  1:     10,     18,       9,       1;
  2:     26,     64,      51,      14,       1;
  3:     76,    230,     250,     115,      20,       1;
  4:    232,    846,    1170,     755,     225,      27,      1;
  5:    764,   3220,    5376,    4480,    1925,     399,     35,      1;
  6:   2620,  12608,   24724,   25256,   14245,    4340,    658,     44,     1;
  7:   9496,  51084,  114696,  138684,   97146,   39501,   8904,   1026,    54,    1;
  8:  35696, 212860,  539100,  753180,  630840,  322497,  98385,  16950,  1530,   65,  1;
  9: 140152, 914936, 2575540, 4079460, 3978150, 2456454, 953337, 224565, 30360, 2200, 77, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(n-1-x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 5*x + x^2.

%t Mathematica

P[0, x_] := 4 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[-(n - 1 - x)*P[n - 1, x] + (2 + 2*x)*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] = 4 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(n - 1 - x)*v[i - 1] + (2 + 2*x)*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 (−∞, -1]
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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by 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 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
Ore-algebra identity of two definitions polytool ? step › direct check
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 (above)

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