A121314 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-9 - 7\,x + 8\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-5 - 9\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	2	3	1													6
3	5	10	6	1												22
4	13	33	29	10	1											86
5	34	105	122	65	15	1										342
6	89	324	468	338	125	21	1									1366
7	233	977	1686	1536	784	217	28	1								5462
8	610	2895	5807	6382	4155	1610	350	36	1							21846
9	1597	8462	19338	24875	19687	9819	3024	534	45	1						87382
10	4181	24465	62731	92410	86085	52529	20985	5304	780	55	1					349526
11	10946	70101	199264	330679	354300	255689	125664	41481	8811	1100	66	1				1398102
12	28657	199368	622152	1148228	1390622	1158558	676242	275968	77022	14003	1507	78	1			5592406
13	75025	563425	1914780	3889680	5253400	4961950	3354500	1631890	565565	135850	21450	2015	91	1		22369622
14	196418	1583643	5821645	12906700	19230330	20304620	15603420	8817660	3655275	1094665	229515	31850	2639	105	1	89478486

Other recurrences

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

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

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

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

Technical details

Node id: n-bfcec383f3588ef6 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-fc5eed90805fc122)
Fingerprint: f6-f1e924027c4d6a7d

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:    1,    1;
  2:    2,    3,     1;
  3:    5,   10,     6,     1;
  4:   13,   33,    29,    10,     1;
  5:   34,  105,   122,    65,    15,    1;
  6:   89,  324,   468,   338,   125,   21,    1;
  7:  233,  977,  1686,  1536,   784,  217,   28,   1;
  8:  610, 2895,  5807,  6382,  4155, 1610,  350,  36,  1;
  9: 1597, 8462, 19338, 24875, 19687, 9819, 3024, 534, 45, 1;

%F Formula

The row polynomials satisfy (2*n-3)*P(n,x) = (8*n-9+(6*n-7)*x)*P(n-1,x) - (5+9*x+4*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[((8*n - 9 + (6*n - 7)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = ((8*n - 9 + (6*n - 7)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] w…?
  - 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 T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] w…?
  - ↑ 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