A008299 → 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	4	3							7
3	11	10							21
4	41	55	15						111
5	162	266	105						533
6	715	1414	805	105					3039
7	3425	7862	5698	1260					18245
8	17722	46281	40845	13230	945				119023
9	98253	287122	297825	126280	17325				826805
10	580317	1873927	2230701	1169245	242550	10395			6107135
11	3633280	12833756	17219202	10736440	2987985	270270			47680933
12	24011157	92011088	137227090	99096998	34644610	4909905	135135		392035983
13	166888165	689014860	1129713676	926345420	389914525	75885810	4729725		3382492181
14	1216070380	5377790747	9606749958	8809901100	4335125795	1077010935	108783675	2027025	30533459615

Technical details

Node id: n-2823757fc1af4ea3 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d9ebe222dad4aaff

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:     4,      3;
  3:    11,     10;
  4:    41,     55,     15;
  5:   162,    266,    105;
  6:   715,   1414,    805,    105;
  7:  3425,   7862,   5698,   1260;
  8: 17722,  46281,  40845,  13230,   945;
  9: 98253, 287122, 297825, 126280, 17325;

%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 T(n,k) of associated Stirling numbers of second kind, n >= 2, 1 <= 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