A166341 → D

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 10 + 2\,x\)

\(P_{2}(x) = 23 + 46\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	10	2															12
2	23	46	3														72
3	50	276	150	4													480
4	105	1228	1842	420	5												3600
5	216	4734	14736	9468	1080	6											30240
6	439	16794	94449	125932	41985	2634	7										282240
7	886	56528	528942	1259320	881570	169584	6202	8									2903040
8	1781	183760	2709312	10528136	13160170	5418624	643160	14248	9								32659200
9	3572	583338	13042368	77927592	157922040	116891388	30432192	2333352	32148	10							399168000
10	7155	1821454	60044535	528756552	1629398670	1955278404	925323966	160118760	8196543	71550	11						5269017600
11	14322	5621772	267499950	3364317180	15063957540	27373897656	21089540556	6728634360	802499850	28108860	157542	12					74724249600
12	28657	17209044	1162759038	20388251900	128232313515	336231275976	392269821972	205171701624	45873566775	3875863460	94649742	343884	13				1133317785600
13	57328	52372214	4960798944	118995058724	1024247029840	3738527917794	6276317151552	4984703890392	1843644653712	297487646810	18189596128	314233284	745264	14			18307441152000
14	114671	158721826	20864453949	674348076724	7781664236495	38460659782638	89465080349253	102245806113432	57690989673957	15563328472990	1854457210991	83457815796	1031691869	1605394	15		313841848320000
15	229358	479605608	86790974142	3733390918736	56805206569830	371942586400200	1164428340258134	1840424510041776	1497122151760458	619904310667000	124971454453626	11200172756208	376094221282	3357239256	3440370	16	5690998849536000

Technical details

Node id: n-b1a3447bb8179bb6 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-0b93002f24124cd4

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:   10,       2;
   2:   23,      46,        3;
   3:   50,     276,      150,         4;
   4:  105,    1228,     1842,       420,          5;
   5:  216,    4734,    14736,      9468,       1080,          6;
   6:  439,   16794,    94449,    125932,      41985,       2634,         7;
   7:  886,   56528,   528942,   1259320,     881570,     169584,      6202,         8;
   8: 1781,  183760,  2709312,  10528136,   13160170,    5418624,    643160,     14248,       9;
   9: 3572,  583338, 13042368,  77927592,  157922040,  116891388,  30432192,   2333352,   32148,    10;
  10: 7155, 1821454, 60044535, 528756552, 1629398670, 1955278404, 925323966, 160118760, 8196543, 71550, 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) = 10 + 2*x, P(2,x) = 23 + 46*x + 3*x^2.

%t Mathematica

P[1, x_] := 10 + 2*x;
P[2, x_] := 23 + 46*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] = 10 + 2*x; v[2] = 23 + 46*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.