A128813 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 12\,n x - n^{2} - 6\,n^{2} x - 9\,n^{2} x^{2} - 3\,n^{3} - 6\,n^{3} x + 9\,n^{3} x^{2} + n^{4} + 6\,n^{4} x + 9\,n^{4} x^{2} - n^{5} - 6\,n^{5} x - 9\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(4\,n - 12\,n x + 4\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	3							8
2	38	84	27						149
3	410	1812	2187	486					4895
4	6190	42852	100287	85536	14580				249445
5	126760	1215012	4371687	7106211	4738500	656100			18214270
6	3409648	42081096	207313272	512028216	643792293	354294000	41334300		1804252825
7	117093680	1774958760	11180204712	37523468808	71117855253	73281287412	34431471900	3472081200	232898421725

All 15 rows

Technical details

Node id: n-27cca71ec864560f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f5f6de77b771b81b

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:            5,             3;
  2:           38,            84,             27;
  3:          410,          1812,           2187,             486;
  4:         6190,         42852,         100287,           85536,           14580;
  5:       126760,       1215012,        4371687,         7106211,         4738500,           656100;
  6:      3409648,      42081096,      207313272,       512028216,       643792293,        354294000,         41334300;
  7:    117093680,    1774958760,    11180204712,     37523468808,     71117855253,      73281287412,      34431471900,       3472081200;
  8:   5011583984,   90286591176,   695227393656,   2968363066680,   7603895780469,   11744281574064,   10282049078796,    4222050739200,    374984769600;
  9: 261942062624, 5474901670416, 49940917502076, 259761709396980, 843350145857289, 1753499683652139, 2291575805392716, 1747284258264660, 637849093089600, 50622943896000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(6*n^5-6*n^4+6*n^3+6*n^2-12*n)*x+(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n-2,x) + (4*n^3+4*n+(12*n^3-12*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (6*n^5 - 6*n^4 + 6*n^3 + 6*n^2 - 12*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (12*n^3 - 12*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] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (6*n^5 - 6*n^4 + 6*n^3 + 6*n^2 - 12*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (12*n^3 - 12*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_A013610 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.