A209414 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 4\,n\right)\,P_{n}(x) = \left(-21 - 23\,x + 28\,n + 32\,n x - 5\,n^{2} - 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-18 - 55\,x - 36\,x^{2} + 9\,n + 26\,n x + 16\,n x^{2}\right)\,P_{n-1}'(x) + \left(-9\,x - 17\,x^{2} - 8\,x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	6	6	1													13
3	18	28	12	1												59
4	54	114	80	20	1											269
5	162	432	422	180	30	1										1227
6	486	1566	1962	1190	350	42	1									5597
7	1458	5508	8424	6640	2828	616	56	1								25531
8	4374	18954	34236	33240	18612	5964	1008	72	1							116461
9	13122	64152	133650	154260	107252	45648	11508	1560	90	1						531243
10	39366	214326	505926	676890	562800	299596	101244	20724	2310	110	1					2423293
11	118098	708588	1869156	2844720	2754270	1762876	750024	207504	35310	3300	132	1				11053979
12	354294	2322594	6770952	11552220	12769434	9557478	4913428	1721720	398970	57486	4576	156	1			50423309
13	1062882	7558272	24131358	45620820	56698866	48612312	29338094	12479116	3682822	727584	90090	6188	182	1		230008587
14	3188646	24446286	84833730	176031630	243015066	234913014	162911994	81642630	29371550	7427706	1268982	136682	8190	210	1	1049196317

Technical details

Node id: n-e3b686c2ee3ca8b9 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d147b3e76f252408

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,     1;
  2:     6,     6,      1;
  3:    18,    28,     12,      1;
  4:    54,   114,     80,     20,      1;
  5:   162,   432,    422,    180,     30,     1;
  6:   486,  1566,   1962,   1190,    350,    42,     1;
  7:  1458,  5508,   8424,   6640,   2828,   616,    56,    1;
  8:  4374, 18954,  34236,  33240,  18612,  5964,  1008,   72,  1;
  9: 13122, 64152, 133650, 154260, 107252, 45648, 11508, 1560, 90, 1;

%F Formula

The row polynomials satisfy (4*n-3)*P(n,x) = -(5*n^2-28*n+21+(8*n^2-32*n+23)*x)*P(n-1,x) + (9*n-18+(26*n-55)*x+(16*n-36)*x^2)*d/dx P(n-1,x) - (9*x+17*x^2+8*x^3)*d^2/dx^2 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[(-(5*n^2 - 28*n + 21 + (8*n^2 - 32*n + 23)*x)*P[n - 1, x] + (9*n - 18 + (26*n - 55)*x + (16*n - 36)*x^2)*D[P[n - 1, x], x] - (9*x + 17*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(4*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] = (-(5*n^2 - 28*n + 21 + (8*n^2 - 32*n + 23)*x)*v[i - 1] + (9*n - 18 + (26*n - 55)*x + (16*n - 36)*x^2)*deriv(v[i - 1], x) - (9*x + 17*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(4*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 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Candidate equalities ?

A209414 → basis A007318 15 rows; no all-index equality certificate was found

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