A166340 → D

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 8 + 2\,x\)

\(P_{2}(x) = 19 + 38\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	8	2															10
2	19	38	3														60
3	42	228	126	4													400
4	89	1020	1530	356	5												3000
5	184	3950	12240	7900	920	6											25200
6	375	14058	78585	104780	35145	2250	7										235200
7	758	47424	440862	1047800	734770	142272	5306	8									2419200
8	1525	154400	2261400	8765896	10957370	4522800	540400	12200	9								27216000
9	3060	490650	10898400	64935880	131488440	97403820	25429600	1962600	27540	10							332640000
10	6131	1533150	50217375	440929000	1357119590	1628543508	771625750	133913000	6899175	61310	11						4390848000
11	12274	4734332	223866750	2807253500	12552007540	22799609112	17572810556	5614507000	671600250	23671660	135014	12					62270208000
12	24561	14497572	973583478	17021157500	106893647115	280096536168	326779292196	171029835384	38297604375	3245278260	79736646	294732	13				944431488000
13	49136	44131302	4155290208	99385011844	854126351440	3115127142018	5228468675136	4153502856024	1537427432592	248462529610	15236064096	264787812	638768	14			15256200960000
14	98287	133769714	17481721221	563409702548	6491302021495	32055406211406	74536584392277	85184667876888	48083109317109	12982604042990	1549376682007	69926884884	869503141	1376018	15		261534873600000
15	196590	404257752	72736048878	3120065013904	47398957667190	310073030163576	970272414271286	1533324021783984	1247493104063082	516788383605960	104277706867818	9360195041712	315189545138	2829804264	2948850	16	4742499041280000

Technical details

Node id: n-db6db25aed063f51 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-1dd701ce264c10cb

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    8,       2;
   2:   19,      38,        3;
   3:   42,     228,      126,         4;
   4:   89,    1020,     1530,       356,          5;
   5:  184,    3950,    12240,      7900,        920,          6;
   6:  375,   14058,    78585,    104780,      35145,       2250,         7;
   7:  758,   47424,   440862,   1047800,     734770,     142272,      5306,         8;
   8: 1525,  154400,  2261400,   8765896,   10957370,    4522800,    540400,     12200,       9;
   9: 3060,  490650, 10898400,  64935880,  131488440,   97403820,  25429600,   1962600,   27540,    10;
  10: 6131, 1533150, 50217375, 440929000, 1357119590, 1628543508, 771625750, 133913000, 6899175, 61310, 11;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-1)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 8 + 2*x, P(2,x) = 19 + 38*x + 3*x^2.

%t Mathematica

P[1, x_] := 8 + 2*x;
P[2, x_] := 19 + 38*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 8 + 2*x; v[2] = 19 + 38*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (3*n + 1 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (above)

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.