A204420 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 18\,x - 3\,x^{2} + 108\,n - 60\,n x + 8\,n x^{2} - 144\,n^{2} + 56\,n^{2} x - 4\,n^{2} x^{2} + 80\,n^{3} - 16\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 2\,x - 12\,n + 4\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	12	6						18
3	345	270	45					660
4	18480	18060	5040	420				42000
5	1596105	1795500	652050	94500	4725			4142880
6	202951980	252369810	108523800	20997900	1871100	62370		586776960
7	35703612945	47927249370	23261733495	5429724300	657431775	39729690	945945	113020427520

All 15 rows

Technical details

Node id: n-5ee4be014b072953 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ad888253c4f868b7

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

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:                0;
  1:                1;
  2:               12,                6;
  3:              345,              270,               45;
  4:            18480,            18060,             5040,             420;
  5:          1596105,          1795500,           652050,           94500,            4725;
  6:        202951980,        252369810,        108523800,        20997900,         1871100,          62370;
  7:      35703612945,      47927249370,      23261733495,      5429724300,       657431775,       39729690,       945945;
  8:    8307234936000,   11854739496600,    6312253147200,   1686922637400,    249729480000,    20772952200,    908107200,    16216200;
  9: 2470641012464625, 3708465702341400, 2126537016704100, 630598380211800, 108018925764750, 11097864577800, 674439866100, 22329707400, 310134825;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(16*n^3-56*n^2+60*n-18)*x+(4*n^2-8*n+3)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (16*n^3 - 56*n^2 + 60*n - 18)*x + (4*n^2 - 8*n + 3)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (4*n - 2)*x)*P[n - 1, 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (16*n^3 - 56*n^2 + 60*n - 18)*x + (4*n^2 - 8*n + 3)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (4*n - 2)*x)*v[i - 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A007318 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of e…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

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.