A372873 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-11 - 12\,x + 11\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-6 - 14\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	5	9	4													18
3	14	36	30	8												88
4	41	138	169	88	16											452
5	122	510	830	650	240	32										2384
6	365	1827	3723	3921	2220	624	64									12744
7	1094	6384	15652	20776	15974	7000	1568	128								68576
8	3281	21876	62726	100580	98001	58800	20832	3840	256							370192
9	9842	73818	242388	455796	538362	411282	200928	59328	9216	512						2001472
10	29525	246045	910390	1964410	2727665	2533849	1582740	648480	163200	21760	1024					10829088
11	88574	811932	3342482	8139824	12993090	14220492	10831150	5695272	2000064	436480	50688	2048				58612096
12	265721	2657214	12046071	32679084	58956231	74301486	66810745	42945672	19421424	5944576	1140480	116736	4096			317289536
13	797162	8635926	42747874	127836046	257213294	366858466	380439670	289239834	160276688	63373024	17132544	2921984	266240	8192		1717746944
14	2391485	27900663	149733605	489347075	1086390795	1729972629	2032824983	1784999841	1171939132	569032464	199303104	48106240	7361536	602112	16384	9299922048

Other recurrences

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

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

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

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

Technical details

Node id: n-b8c7876a249ffefc · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-6aae761843f53e9e)
Fingerprint: f6-c654e1ec9ab2ab59

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:    5,     9,      4;
  3:   14,    36,     30,      8;
  4:   41,   138,    169,     88,     16;
  5:  122,   510,    830,    650,    240,     32;
  6:  365,  1827,   3723,   3921,   2220,    624,     64;
  7: 1094,  6384,  15652,  20776,  15974,   7000,   1568,   128;
  8: 3281, 21876,  62726, 100580,  98001,  58800,  20832,  3840,  256;
  9: 9842, 73818, 242388, 455796, 538362, 411282, 200928, 59328, 9216, 512;

%F Formula

The row polynomials satisfy (2*n-2)*P(n,x) = (11*n-11+(12*n-12)*x)*P(n-1,x) - (6+14*x+8*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n - 11 + (12*n - 12)*x)*P[n - 1, x] - (6 + 14*x + 8*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((11*n - 11 + (12*n - 12)*x)*v[i - 1] - (6 + 14*x + 8*x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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 flattened Catalan words of length n with exactly k runs of descents.?
  - 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 flattened Catalan words of length n with exactly k runs of descents.?
  - ↑ 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