A112007 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	8	2													10
3	58	44	3												105
4	444	656	156	4											1260
5	3708	8800	4356	456	5										17325
6	33984	116280	96360	22440	1200	6									270270
7	341136	1570608	1932060	783200	99750	2964	7								4729725
8	3733920	22052592	37320192	23062000	5312500	403560	7028	8							91891800
9	44339040	324373824	716714712	621429536	223825000	31956960	1531936	16208	9						1964187225
10	568356480	5014962432	13926491856	16010780352	8241921520	1886218320	176707328	5560304	36648	10					45831035250
11	7827719040	81576603648	276599370672	404721732096	280206993920	93293764704	14360379432	919011008	19535634	81660	11				1159525191825
12	115336085760	1395858873600	5649238985472	10199461893696	9090943211520	4126320278304	940263941568	101358542208	4565333484	66996960	179916	12			31623414322500
13	1810992556800	25101808035840	119068734167424	258913500559488	287471867322960	169799460396672	53443825045744	8677412099328	675144470052	21898298080	225587868	392856	13		924984868933125
14	30196376985600	473816543086080	2595069759658752	6665697945085824	8985856103452800	6670484835418080	2760559114708640	627131078650496	74775998951712	4298260063400	102217796120	748894680	851552	14	28887988983603750

Technical details

Node id: n-5d6fe78c6ecdb1ab · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3dcce632d1612cce

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:        8,         2;
  3:       58,        44,         3;
  4:      444,       656,       156,         4;
  5:     3708,      8800,      4356,       456,         5;
  6:    33984,    116280,     96360,     22440,      1200,        6;
  7:   341136,   1570608,   1932060,    783200,     99750,     2964,       7;
  8:  3733920,  22052592,  37320192,  23062000,   5312500,   403560,    7028,     8;
  9: 44339040, 324373824, 716714712, 621429536, 223825000, 31956960, 1531936, 16208, 9;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(n^3+n^2+(2*n^3-4*n^2)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (n^2*x-n*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2+n-2+(2*n^2-4*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx 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[Together[(-(n^3 + n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (n^2*x - n*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 + n - 2 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + n^2 + (2*n^3 - 4*n^2)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (n^2*x - n*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 + n - 2 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((n - 1)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(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
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 Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Second-order Eulerian triangle T(n,k), 1 <= k <= n.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A163936 → D

References

[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.