A099093 → D

nonnegative coefficients

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

Recurrence

\(\left(3\,x - n - 3\,n x + n^{2}\right)\,P_{n}(x) = \left(-6 - 21\,x - 18\,x^{2} + 8\,n + 15\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 15\,x^{2} - 9\,x^{3} + 4\,n x + 3\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3														3
2	3	18													21
3	0	36	81												117
4	0	18	243	324											585
5	0	0	243	1296	1215										2754
6	0	0	81	1944	6075	4374									12474
7	0	0	0	1296	12150	26244	15309								54999
8	0	0	0	324	12150	65610	107163	52488							237735
9	0	0	0	0	6075	87480	321489	419904	177147						1012095
10	0	0	0	0	1215	65610	535815	1469664	1594323	590490					4257117
11	0	0	0	0	0	26244	535815	2939328	6377292	5904900	1948617				17732196
12	0	0	0	0	0	4374	321489	3674160	14880348	26572050	21434787	6377292			73264500
13	0	0	0	0	0	0	107163	2939328	22320522	70858800	107173935	76527504	20726199		300653451
14	0	0	0	0	0	0	15309	1469664	22320522	124002900	321521805	420901272	269440587	66961566	1226633625

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Technical details

Node id: n-3d0a61f38b990a78 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-38e17b131fbaf7c5

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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 ?

No OEIS matches.

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: 0;
  1: 3;
  2: 3, 18;
  3: 0, 36,  81;
  4: 0, 18, 243,  324;
  5: 0,  0, 243, 1296,  1215;
  6: 0,  0,  81, 1944,  6075,  4374;
  7: 0,  0,   0, 1296, 12150, 26244,  15309;
  8: 0,  0,   0,  324, 12150, 65610, 107163,  52488;
  9: 0,  0,   0,    0,  6075, 87480, 321489, 419904, 177147;

%F Formula

The row polynomials satisfy (n^2-n-(3*n-3)*x)*P(n,x) = -(2*n^2-8*n+6-(15*n-21)*x+18*x^2)*P(n-1,x) + ((4*n-4)*x+(3*n-15)*x^2-9*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 8*n + 6 - (15*n - 21)*x + 18*x^2)*P[n - 1, x] + ((4*n - 4)*x + (3*n - 15)*x^2 - 9*x^3)*D[P[n - 1, x], x])/(n^2 - n - (3*n - 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 8*n + 6 - (15*n - 21)*x + 18*x^2)*v[i - 1] + ((4*n - 4)*x + (3*n - 15)*x^2 - 9*x^3)*deriv(v[i - 1], x))/(n^2 - n - (3*n - 3)*x)); 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-73ab4d345aa74eb4
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-73ab4d345aa74eb4
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-73ab4d345aa74eb4

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 derivative_polynomial, shift 0 e-73ab4d345aa74eb4

Laguerre’s theorem on differential operators [SP14] 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 ?
    
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    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 ?
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    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 ?

No candidate equalities or related sequences.

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.