A272867 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	4	2													6
2	8	36	24	4											72
3	12	102	264	204	60	6									648
4	16	200	912	1816	1520	600	112	8							5184
5	20	330	2160	7080	12120	10620	5040	1320	180	10					38880
6	24	492	4200	19260	51120	79416	71568	38520	12600	2460	264	12			279936
7	28	686	7224	42644	154420	353682	513744	471576	277956	106610	26488	4116	364	14	1959552

All 15 rows

Technical details

Node id: n-749edefcbd70f36b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-70b67aafe0c2287c

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  4,    2;
  2:  8,   36,    24,      4;
  3: 12,  102,   264,    204,     60,       6;
  4: 16,  200,   912,   1816,   1520,     600,     112,        8;
  5: 20,  330,  2160,   7080,  12120,   10620,    5040,     1320,      180,       10;
  6: 24,  492,  4200,  19260,  51120,   79416,   71568,    38520,    12600,     2460,      264,      12;
  7: 28,  686,  7224,  42644, 154420,  353682,  513744,   471576,   277956,   106610,    26488,    4116,     364,     14;
  8: 32,  912, 11424,  82544, 379680, 1158864, 2380448,  3291696,  3060576,  1931440,   835296,  247632,   49504,   6384,   480,   16;
  9: 36, 1170, 16992, 145296, 811440, 3105144, 8320032, 15718896, 20933784, 19648620, 13074336, 6210288, 2109744, 508536, 84960, 9360, 612, 18;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n+4*n*x+n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 4*n*x + n*x^2)*P[n - 1, x])/(n - 1)]];
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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 4*n*x + n*x^2)*v[i - 1])/(n - 1)); 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.