A272867 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-3 + 2\,x + x^{2} + 4\,n + 2\,n x\right)\,P_{n}(x) = \left(1 + 8\,x + 18\,x^{2} + 8\,x^{3} + x^{4} + 4\,n + 18\,n x + 12\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	6	1													12
2	9	44	42	12	1											108
3	13	114	315	292	111	18	1									864
4	17	216	1012	2120	1974	904	212	24	1							6480
5	21	350	2325	7800	13890	13044	6810	2040	345	30	1					46656
6	25	516	4446	20660	55935	89640	84804	48744	17415	3860	510	36	1			326592
7	29	714	7567	45052	165081	384566	572691	544968	336903	137494	37149	6524	707	42	1	2239488

All 15 rows

Technical details

Node id: n-27f8d8257df8b7b5 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-732b3fedf81b90c1

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:  5,    6,     1;
  2:  9,   44,    42,     12,      1;
  3: 13,  114,   315,    292,    111,      18,       1;
  4: 17,  216,  1012,   2120,   1974,     904,     212,       24,        1;
  5: 21,  350,  2325,   7800,  13890,   13044,    6810,     2040,      345,       30,        1;
  6: 25,  516,  4446,  20660,  55935,   89640,   84804,    48744,    17415,     3860,      510,      36,       1;
  7: 29,  714,  7567,  45052, 165081,  384566,  572691,   544968,   336903,   137494,    37149,    6524,     707,     42,      1;
  8: 33,  944, 11880,  86352, 400316, 1234800, 2573592,  3631760,  3472038,  2271504,  1028440,  323568,   70140,  10192,    936,    48,    1;
  9: 37, 1206, 17577, 150960, 847764, 3267432, 8837556, 16907472, 22898646, 21974596, 15039198, 7398864, 2627268, 670824, 121284, 15024, 1197, 54, 1;

%F Formula

The row polynomials satisfy (4*n-3+(2*n+2)*x+x^2)*P(n,x) = (4*n+1+(18*n+8)*x+(12*n+18)*x^2+(2*n+8)*x^3+x^4)*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[((4*n + 1 + (18*n + 8)*x + (12*n + 18)*x^2 + (2*n + 8)*x^3 + x^4)*P[n - 1, x])/(4*n - 3 + (2*n + 2)*x + x^2)]];
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] = ((4*n + 1 + (18*n + 8)*x + (12*n + 18)*x^2 + (2*n + 8)*x^3 + x^4)*v[i - 1])/(4*n - 3 + (2*n + 2)*x + x^2)); 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.