A166344 → D

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 6 + 2\,x\)

\(P_{2}(x) = 15 + 30\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	6	2															8
2	15	30	3														48
3	34	180	102	4													320
4	73	812	1218	292	5												2400
5	152	3166	9744	6332	760	6											20160
6	311	11322	62721	83628	28305	1866	7										188160
7	630	38320	352782	836280	587970	114960	4410	8									1935360
8	1269	125040	1813488	7003656	8754570	3626976	437640	10152	9								21772800
9	2548	397962	8754432	51944168	105054840	77916252	20427008	1591848	22932	10							266112000
10	5107	1244846	40390215	353101448	1084840510	1301808612	617927534	107707240	5601807	51070	11						3512678400
11	10226	3846892	180233550	2250189820	10040057540	18225320568	14056080556	4500379640	540700650	19234460	112486	12					49816166400
12	20465	11786100	784407918	13654063100	85554980715	223961796360	261288762420	136887969144	30721641975	2614693060	64823550	245580	13				755545190400
13	40944	35890390	3349781472	79774964964	684005673040	2491726366242	4180620198720	3322301821656	1231210211472	199437412410	12282532064	215342340	532272	14			12204960768000
14	81903	108817602	14098988493	452471328372	5200939806495	25650152640174	59608088435301	68123529640344	38475228960261	10401879612990	1244296153023	56395953972	707314413	1146642	15		209227898880000
15	163822	328909896	58681123614	2506739109072	37992708764550	248203473926952	776116488284438	1226223533526192	997864056365706	413672456544920	83583959282010	7520217327216	254284868994	2302369272	2457330	16	3793999233024000

Technical details

Node id: n-415ba9adacda5f2f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-563234e7acaf8cdf

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:    6,       2;
   2:   15,      30,        3;
   3:   34,     180,      102,         4;
   4:   73,     812,     1218,       292,          5;
   5:  152,    3166,     9744,      6332,        760,          6;
   6:  311,   11322,    62721,     83628,      28305,       1866,         7;
   7:  630,   38320,   352782,    836280,     587970,     114960,      4410,         8;
   8: 1269,  125040,  1813488,   7003656,    8754570,    3626976,    437640,     10152,       9;
   9: 2548,  397962,  8754432,  51944168,  105054840,   77916252,  20427008,   1591848,   22932,    10;
  10: 5107, 1244846, 40390215, 353101448, 1084840510, 1301808612, 617927534, 107707240, 5601807, 51070, 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) = 6 + 2*x, P(2,x) = 15 + 30*x + 3*x^2.

%t Mathematica

P[1, x_] := 6 + 2*x;
P[2, x_] := 15 + 30*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] = 6 + 2*x; v[2] = 15 + 30*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
  - 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 polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ 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.