A306364 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	15	3							18
3	105	45							150
4	945	630	45						1620
5	10395	9450	1575						21420
6	135135	155925	42525	1575					335160
7	2027025	2837835	1091475	99225					6055560
8	34459425	56756700	28378350	4365900	99225				124059600
9	654729075	1240539300	766215450	170270100	9823275				2841577200
10	13749310575	29462808375	21709437750	6385128750	638512875	9823275			71955021600
11	316234143225	756212081625	648181784250	238803815250	35118208125	1404728325			1995954760800
12	7905853580625	20871453452850	20417726203875	9074544979500	1791028614375	126425549250	1404728325		60188437108800
13	213458046676875	616656579288750	678322237217625	353907254200500	88476813550125	9313348794750	273922023375		1960408201752000
14	6190283353629375	19424682247595625	23741278302616875	14244766981570125	4335363863956125	619337694850875	32596720781625	273922023375	68588583087024000

Other recurrences

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

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

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

\(P_{1}(x) = 3\)

Technical details

Node id: n-bab8f81b674979eb · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-5344783b25882878)
Fingerprint: f6-803a5d83f859558a

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:         3;
  2:        15,          3;
  3:       105,         45;
  4:       945,        630,        45;
  5:     10395,       9450,      1575;
  6:    135135,     155925,     42525,      1575;
  7:   2027025,    2837835,   1091475,     99225;
  8:  34459425,   56756700,  28378350,   4365900,   99225;
  9: 654729075, 1240539300, 766215450, 170270100, 9823275;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 1 + (n - 1)*x)*P[n - 1, x] - (2*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; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + (n - 1)*x)*v[i - 1] - (2*x + 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 (−∞, -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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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 Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 …?
  - 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 Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 …?
  - ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A008556 → D

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