A257180 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-36 - 18\,x - 6\,x^{2} - 12\,n - 24\,n x - 11\,n x^{2} - 6\,n^{2} x - 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-9\,x + 3\,x^{2} + 6\,x^{3} - 3\,n x - 2\,n x^{2} + 5\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(17 + 8\,x + 7\,n + 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 24 + 18\,x\)

\(P_{2}(x) = 141 + 282\,x + 81\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
1	24	18								42
2	141	282	81							504
3	726	2820	2178	324						6048
4	3471	22812	34218	13884	1215					75600
5	15828	162654	410616	325308	79140	4374				997920
6	69873	1066518	4171563	5562084	2666295	419238	15309			13970880
7	301362	6590304	37827594	77869176	63045990	19770912	2109534	52488		207567360
8	1277619	38978760	315934668	948578568	1185723210	631869336	136425660	10220952	177147	3269185920

All 15 rows

Technical details

Node id: n-7b43ff23f0ec92e7 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-9c764471e214691a

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:       24,         18;
   2:      141,        282,          81;
   3:      726,       2820,        2178,          324;
   4:     3471,      22812,       34218,        13884,         1215;
   5:    15828,     162654,      410616,       325308,        79140,         4374;
   6:    69873,    1066518,     4171563,      5562084,      2666295,       419238,        15309;
   7:   301362,    6590304,    37827594,     77869176,     63045990,     19770912,      2109534,       52488;
   8:  1277619,   38978760,   315934668,    948578568,   1185723210,    631869336,    136425660,    10220952,     177147;
   9:  5346672,  223001418,  2480289408,  10431265992,  18971571360,  15646898988,   5787341952,   892005672,   48120048,    590490;
  10: 22154325, 1243327218, 18561259845, 106089387384, 269124313290, 322949175948, 185656427922, 49496692920, 5594972481, 221543250, 1948617;

%F Formula

The row polynomials satisfy (n+2)*P(n,x) = -(12*n+36+(6*n^2+24*n+18)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((3*n+9)*x+(n^2+2*n-3)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (7*n+17+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 24 + 18*x, P(2,x) = 141 + 282*x + 81*x^2.

%t Mathematica

P[1, x_] := 24 + 18*x;
P[2, x_] := 141 + 282*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 36 + (6*n^2 + 24*n + 18)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((3*n + 9)*x + (n^2 + 2*n - 3)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (7*n + 17 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 24 + 18*x; v[2] = 141 + 282*x + 81*x^2; for(n = 3, N, my(i = n); v[i] = (-(12*n + 36 + (6*n^2 + 24*n + 18)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((3*n + 9)*x + (n^2 + 2*n - 3)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (7*n + 17 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 1], x))/(n + 2)); 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, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.?
    
    ↑ Liu–Wang theorem (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.