A124324 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	1							3
3	5	4							9
4	15	17	3						35
5	52	76	25						153
6	203	362	175	15					755
7	877	1842	1176	210					4105
8	4140	9991	7917	2170	105				24323
9	21147	57568	54405	20188	2205				155513
10	115975	351125	384951	180355	31500	945			1064851
11	678570	2259302	2815648	1593350	386155	27720			7760745
12	4213597	15288000	21323830	14136386	4399780	523215	10395		59895203
13	27644437	108478124	167280048	127052640	48369035	8168160	405405		487397849
14	190899322	805037105	1358951854	1162669508	523656133	115440325	9774765	135135	4166564147

Technical details

Node id: n-fc2cd21d94561523 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-ff7155220b16c98b

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:     1;
  2:     2,     1;
  3:     5,     4;
  4:    15,    17,     3;
  5:    52,    76,    25;
  6:   203,   362,   175,    15;
  7:   877,  1842,  1176,   210;
  8:  4140,  9991,  7917,  2170,  105;
  9: 21147, 57568, 54405, 20188, 2205;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n - 1 + (n - 1)*x)*P[n - 2, x] + P[n - 1, x] + (1 + 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] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + (n - 1)*x)*v[i - 2] + v[i - 1] + (1 + 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 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 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=k<=floor(n/2)).?
  - 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