A099093 → reverse

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	9	3							12
3	27	18							45
4	81	81	9						171
5	243	324	81						648
6	729	1215	486	27					2457
7	2187	4374	2430	324					9315
8	6561	15309	10935	2430	81				35316
9	19683	52488	45927	14580	1215				133893
10	59049	177147	183708	76545	10935	243			507627
11	177147	590490	708588	367416	76545	4374			1924560
12	531441	1948617	2657205	1653372	459270	45927	729		7296561
13	1594323	6377292	9743085	7085880	2480058	367416	15309		27663363
14	4782969	20726199	35075106	29229255	12400290	2480058	183708	2187	104879772

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Technical details

Node id: n-0515531ee2138839 · revision 1 · JSON
Definition source: derived:reversal
Fingerprint: f6-62866b7caf28db73

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

OEIS matches ?

A030195: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     3;
  2:     9,     3;
  3:    27,    18;
  4:    81,    81,     9;
  5:   243,   324,    81;
  6:   729,  1215,   486,    27;
  7:  2187,  4374,  2430,   324;
  8:  6561, 15309, 10935,  2430,   81;
  9: 19683, 52488, 45927, 14580, 1215;

%F Formula

The row polynomials satisfy n*P(n,x) = (3*n+(2*n-2)*x)*P(n-1,x) - (3*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[((3*n + (2*n - 2)*x)*P[n - 1, x] - (3*x + 4*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] = ((3*n + (2*n - 2)*x)*v[i - 1] - (3*x + 4*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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-450f652a77090025
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-450f652a77090025
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-450f652a77090025

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image reversal, shift 0 e-450f652a77090025

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 3+3x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - basis_map A099093 e-97db6227df9d6f92
    
    Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?
    
    premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    
    no recorded property covers this requirement
  - theorem_check ma_wang_interval e-a3711e6f8037cbc8 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?
  - basis_map A027466 e-04516916870a85c6
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 7]
    
    preserver_image affine, shift 0 e-bc143c20b95292bc
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A038243 e-089b35b0d9fc7c83
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 5]
    
    preserver_image affine, shift 0 e-ae28d4484158822f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A007318 e-344ac1c74a22d15f
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    
    preserver_image affine, shift 0 e-9ac4433d08f91ae9
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A038231 e-3547903a43c489fd
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 4]
    
    preserver_image affine, shift 0 e-4e1348328210bdec
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A027467 e-35fdacf74fcfdf8c
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 15]
    
    preserver_image affine, shift 0 e-b320272c2500474a
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A027465 e-69bbc7837aabb73c
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 3]
    
    preserver_image affine, shift 0 e-7438d71d24adbb8f
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A038291 e-70271c52b20af48c
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 9]
    
    preserver_image affine, shift 0 e-7d49a73cf4bf68ad
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A038207 e-726dbe4e92e67f96
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 2]
    
    preserver_image affine, shift 0 e-8e25fd3c0a02e03b
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A038279 e-78bb889c5d4c7dc1
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-b4aa148b93b9dcb2
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A038303 e-a4925a412f743bbb
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 10]
    
    preserver_image affine, shift 0 e-d18687645e4e9661
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A038327 e-e0b7b31567b585e1
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 12]
    
    preserver_image affine, shift 0 e-2a73ae63953e9ff6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)
  - basis_map A038255 e-e9cb24f9617f2dae
    
    Geometric basis map (elementary) polytool ?
    
    premise (sequence): basis-map coefficients of Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 6]
    
    preserver_image affine, shift 0 e-6c0da882dbe0cffe
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Riordan array (1, 3+3x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
    
    ↵ theorem_check ma_wang_interval e-a3711e6f8037cbc8 (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A099093: Riordan array (1, 3+3x).

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741