A128813 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 8\,n x - n^{2} - 4\,n^{2} x - 4\,n^{2} x^{2} - 3\,n^{3} - 4\,n^{3} x + 4\,n^{3} x^{2} + n^{4} + 4\,n^{4} x + 4\,n^{4} x^{2} - n^{5} - 4\,n^{5} x - 4\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(4\,n - 8\,n x + 4\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	28	44	12						84
3	292	884	756	144					2076
4	4332	20284	32476	19584	2880				79556
5	87792	565984	1376656	1529544	720000	86400			4366376
6	2345376	19412224	64306768	107326368	92176848	35769600	3628800		324965984
7	80158496	813494016	3435923856	7752722336	9926204688	6983042688	2311545600	203212800	31506304480

All 15 rows

Technical details

Node id: n-78fe0e73e916e68a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6aa6dbddb3aeaca3

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 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:            4,             2;
  2:           28,            44,             12;
  3:          292,           884,            756,            144;
  4:         4332,         20284,          32476,          19584,            2880;
  5:        87792,        565984,        1376656,        1529544,          720000,           86400;
  6:      2345376,      19412224,       64306768,      107326368,        92176848,        35769600,         3628800;
  7:     80158496,     813494016,     3435923856,     7752722336,      9926204688,      6983042688,      2311545600,      203212800;
  8:   3418609568,   41189356800,   212327203152,   607823932832,   1046617911504,   1091766198528,    652318292736,   188581478400,    14631321600;
  9: 178195506368, 2489120668800, 15184246815552, 52869114168992, 115074675129024, 160810760125728, 141909775151616, 73821175856640, 18962192793600, 1316818944000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(4*n^5-4*n^4+4*n^3+4*n^2-8*n)*x+(4*n^5-4*n^4-4*n^3+4*n^2)*x^2)*P(n-2,x) + (4*n^3+4*n+(8*n^3-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (8*n^3 - 8*n)*x)*P[n - 1, x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (8*n^3 - 8*n)*x)*v[i - 1])/(4*n - 4)); 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_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.