A099097 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - 24\,x + 96\,x^{2} - 27\,x^{3} + 15\,n + 39\,n x - 123\,n x^{2} + 27\,n x^{3} - 12\,n^{2} - 15\,n^{2} x + 27\,n^{2} x^{2} + 3\,n^{3}\right)\,P_{n}(x) = \left(16 + 68\,x - 20\,x^{2} - 171\,x^{3} + 81\,x^{4} - 28\,n - 117\,n x - 26\,n x^{2} + 63\,n x^{3} + 14\,n^{2} + 37\,n^{2} x - 18\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x + 30\,x^{2} - 29\,x^{3} - 90\,x^{4} + 81\,x^{5} - 12\,n x - 47\,n x^{2} - 9\,n x^{3} + 81\,n x^{4} + 4\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	3														6
3	1	19	9													29
4	0	12	87	27												126
5	0	2	82	351	81											516
6	0	0	27	441	1323	243										2034
7	0	0	3	217	2079	4779	729									7807
8	0	0	0	48	1362	9018	16767	2187								29382
9	0	0	0	4	451	7380	36936	57591	6561							108923
10	0	0	0	0	75	3255	36261	145071	194643	19683						398988
11	0	0	0	0	5	811	19980	166131	551853	649539	59049					1447368
12	0	0	0	0	0	108	6633	109809	722196	2047032	2145447	177147				5208372
13	0	0	0	0	0	6	1324	45549	556794	3013686	7440174	7026831	531441			18615805
14	0	0	0	0	0	0	147	12117	277074	2655018	12170655	26591733	22851963	1594323		66153030
15	0	0	0	0	0	0	7	2017	92106	1541106	12061305	47849373	93710763	73870299	4782969	233909945

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Technical details

Node id: n-84ea1b0f8a235f9e · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-1e8466a1ae3eeaa5

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 3,  3;
   3: 1, 19,  9;
   4: 0, 12, 87,  27;
   5: 0,  2, 82, 351,   81;
   6: 0,  0, 27, 441, 1323,  243;
   7: 0,  0,  3, 217, 2079, 4779,   729;
   8: 0,  0,  0,  48, 1362, 9018, 16767,   2187;
   9: 0,  0,  0,   4,  451, 7380, 36936,  57591,   6561;
  10: 0,  0,  0,   0,   75, 3255, 36261, 145071, 194643, 19683;

%F Formula

The row polynomials satisfy (3*n^3-12*n^2+15*n-6-(15*n^2-39*n+24)*x+(27*n^2-123*n+96)*x^2+(27*n-27)*x^3)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(37*n^2-117*n+68)*x+(18*n^2+26*n+20)*x^2-(63*n-171)*x^3-81*x^4)*P(n-1,x) + ((4*n^2-12*n+8)*x+(9*n^2-47*n+30)*x^2-(9*n+29)*x^3+(81*n-90)*x^4+81*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 14*n^2 + 28*n - 16 - (37*n^2 - 117*n + 68)*x + (18*n^2 + 26*n + 20)*x^2 - (63*n - 171)*x^3 - 81*x^4)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (9*n^2 - 47*n + 30)*x^2 - (9*n + 29)*x^3 + (81*n - 90)*x^4 + 81*x^5)*D[P[n - 1, x], x])/(3*n^3 - 12*n^2 + 15*n - 6 - (15*n^2 - 39*n + 24)*x + (27*n^2 - 123*n + 96)*x^2 + (27*n - 27)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 14*n^2 + 28*n - 16 - (37*n^2 - 117*n + 68)*x + (18*n^2 + 26*n + 20)*x^2 - (63*n - 171)*x^3 - 81*x^4)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (9*n^2 - 47*n + 30)*x^2 - (9*n + 29)*x^3 + (81*n - 90)*x^4 + 81*x^5)*deriv(v[i - 1], x))/(3*n^3 - 12*n^2 + 15*n - 6 - (15*n^2 - 39*n + 24)*x + (27*n^2 - 123*n + 96)*x^2 + (27*n - 27)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-a3e4828d005b5462
Coefficients of one sign. Every nonzero row, n >= 1: 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-a3e4828d005b5462
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-a3e4828d005b5462

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-a3e4828d005b5462

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Riordan array (1, 3+x).
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    Geometric basis map (elementary) polytool ?
    
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    preserver_image affine, shift 0 e-4899b779b1a74e41
    
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  - basis_map A038327 e-e4ffd1425a5524fc
    
    Geometric basis map (elementary) polytool ?
    
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    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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    Geometric basis map (elementary) polytool ?
    
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    preserver_image affine, shift 0 e-7827a087c75bbe15
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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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.