A099097 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-27\,x - 3\,n + 54\,n x - 27\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(18\,x + 4\,n + 9\,n x - 162\,n x^{2} + 2\,n^{2} + 45\,n^{2} x - 2\,n^{3}\right)\,P_{n-1}(x) + \left(4\,n x - 27\,n x^{2} - 81\,n x^{3} + 4\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	12														12
3	0	4	81													85
4	0	0	54	432												486
5	0	0	9	432	2025											2466
6	0	0	0	144	2700	8748										11592
7	0	0	0	16	1350	14580	35721									51667
8	0	0	0	0	300	9720	71442	139968								221430
9	0	0	0	0	25	3240	59535	326592	531441							920833
10	0	0	0	0	0	540	26460	326592	1417176	1968300						3739068
11	0	0	0	0	0	36	6615	181440	1653372	5904900	7144929					14891292
12	0	0	0	0	0	0	882	60480	1102248	7873200	23816430	25509168				58362408
13	0	0	0	0	0	0	49	12096	459270	6123600	35724645	93533616	89813529			225666805
14	0	0	0	0	0	0	0	1344	122472	3061800	31755240	155889360	359254116	312487308		862571640
15	0	0	0	0	0	0	0	64	20412	1020600	18523890	155889360	658632546	1354111668	1076168025	3264366565

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Technical details

Node id: n-aaf403a2fe94169d · revision 1 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-618b55a2927dda81

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: 0, 12;
   3: 0,  4, 81;
   4: 0,  0, 54, 432;
   5: 0,  0,  9, 432, 2025;
   6: 0,  0,  0, 144, 2700,  8748;
   7: 0,  0,  0,  16, 1350, 14580, 35721;
   8: 0,  0,  0,   0,  300,  9720, 71442, 139968;
   9: 0,  0,  0,   0,   25,  3240, 59535, 326592,  531441;
  10: 0,  0,  0,   0,    0,   540, 26460, 326592, 1417176, 1968300;

%F Formula

The row polynomials satisfy (3*n^3-3*n-(27*n^2-54*n+27)*x)*P(n,x) = -(2*n^3-2*n^2-4*n-(45*n^2+9*n+18)*x+162*n*x^2)*P(n-1,x) + ((4*n^2+4*n)*x+(9*n^2-27*n)*x^2-81*n*x^3)*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 - 2*n^2 - 4*n - (45*n^2 + 9*n + 18)*x + 162*n*x^2)*P[n - 1, x] + ((4*n^2 + 4*n)*x + (9*n^2 - 27*n)*x^2 - 81*n*x^3)*D[P[n - 1, x], x])/(3*n^3 - 3*n - (27*n^2 - 54*n + 27)*x)]];
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 - 2*n^2 - 4*n - (45*n^2 + 9*n + 18)*x + 162*n*x^2)*v[i - 1] + ((4*n^2 + 4*n)*x + (9*n^2 - 27*n)*x^2 - 81*n*x^3)*deriv(v[i - 1], x))/(3*n^3 - 3*n - (27*n^2 - 54*n + 27)*x)); 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-ab38a7348371e4e3
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-ab38a7348371e4e3
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-ab38a7348371e4e3

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 euler_polynomial, shift 0 e-ab38a7348371e4e3

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?
- premise (sequence): Riordan array (1, 3+x).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
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    Geometric basis map (elementary) polytool ?
    
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    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
    
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    preserver_image affine, shift 0 e-4899b779b1a74e41
    
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    preserver_image affine, shift 0 e-5d14c25d8e006de4
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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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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    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
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    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
    
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  - basis_map A038231 e-f299bbf6486ea9ca
    
    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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    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 8]
    
    preserver_image affine, shift 0 e-7827a087c75bbe15
    
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    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
    
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Related sequences ?

No candidate equalities or related sequences.

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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.