A119900 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-x + 2\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	4	7	3						14
3	8	20	16	4					48
4	16	52	61	30	5				164
5	32	128	198	146	50	6			560
6	64	304	584	575	301	77	7		1912
7	128	704	1616	1992	1408	560	112	8	6528

All 15 rows

Other recurrences

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

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

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

\(P_{1}(x) = 2 + 2\,x\)

Technical details

Node id: n-7ba9c2ffd0ca797f · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-33a90d5846c87322)
Fingerprint: f6-101d6912c5fd0089

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:   2,    2;
  2:   4,    7,     3;
  3:   8,   20,    16,     4;
  4:  16,   52,    61,    30,     5;
  5:  32,  128,   198,   146,    50,     6;
  6:  64,  304,   584,   575,   301,    77,    7;
  7: 128,  704,  1616,  1992,  1408,   560,  112,    8;
  8: 256, 1600,  4272,  6328,  5641,  3060,  966,  156,   9;
  9: 512, 3584, 10912, 18880, 20330, 14002, 6084, 1572, 210, 10;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n+(3*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[Together[((2*n + (3*n - 1)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];
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 + (3*n - 1)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(n)); 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=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 Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=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