A256890 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,x + n\right)\,P_{n}(x) = \left(-6 - 10\,x + 3\,x^{2} - 3\,x^{3} - 2\,x^{4} - 6\,n - 2\,n x - 9\,n x^{2} - 2\,n x^{3} - 2\,n x^{4} - 4\,n^{2} x - n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 2\,x^{2} + 3\,x^{3} - x^{4} + 2\,x^{5} - 2\,n x + n x^{2} - 2\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 10\,x + x^{2} + 2\,x^{3} + 5\,n + n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 16 + 20\,x + 4\,x^{2}\)

\(P_{2}(x) = 60 + 156\,x + 76\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	16	20	4								40
2	60	156	76	8							300
3	212	1028	1004	260	16						2520
4	716	5972	10576	5380	844	32					23520
5	2340	31636	94000	85160	26060	2660	64				241920
6	7468	156644	735492	1101400	593540	118692	8236	128			2721600
7	23428	737828	5229220	12197404	10763180	3767932	519532	25220	256		33264000
8	72588	3347796	34573192	119890076	163367448	93098572	22443464	2214468	76684	512	439084800

All 15 rows

Technical details

Node id: n-ad4d41d896b8148e · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9f7813ee7004ecb1

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     16,       20,          4;
   2:     60,      156,         76,          8;
   3:    212,     1028,       1004,        260,          16;
   4:    716,     5972,      10576,       5380,         844,          32;
   5:   2340,    31636,      94000,      85160,       26060,        2660,          64;
   6:   7468,   156644,     735492,    1101400,      593540,      118692,        8236,        128;
   7:  23428,   737828,    5229220,   12197404,    10763180,     3767932,      519532,      25220,       256;
   8:  72588,  3347796,   34573192,  119890076,   163367448,    93098572,    22443464,    2214468,     76684,      512;
   9: 222884, 14766260,  216171080, 1074412960,  2157148560,  1888360712,   738625720,  127810560,   9268940,   232100,   1024;
  10: 679916, 63736388, 1294189660, 8956353360, 25523341560, 32745264888, 19614961464, 5500961040, 704650140, 38307940, 700396, 2048;

%F Formula

The row polynomials satisfy (n+2*x)*P(n,x) = -(6*n+6+(4*n^2+2*n+10)*x+(n^3+n^2+9*n-3)*x^2+(3*n^2+2*n+3)*x^3+(2*n+2)*x^4)*P(n-2,x) - ((2*n+2)*x+(n^2-n+2)*x^2-(n^2-2*n+3)*x^3-(3*n-1)*x^4-2*x^5)*d/dx P(n-2,x) + (5*n+2+(2*n^2+n+10)*x+(5*n+1)*x^2+2*x^3)*P(n-1,x) + (n*x-(n-2)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 16 + 20*x + 4*x^2, P(2,x) = 60 + 156*x + 76*x^2 + 8*x^3.

%t Mathematica

P[1, x_] := 16 + 20*x + 4*x^2;
P[2, x_] := 60 + 156*x + 76*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n + 6 + (4*n^2 + 2*n + 10)*x + (n^3 + n^2 + 9*n - 3)*x^2 + (3*n^2 + 2*n + 3)*x^3 + (2*n + 2)*x^4)*P[n - 2, x] - ((2*n + 2)*x + (n^2 - n + 2)*x^2 - (n^2 - 2*n + 3)*x^3 - (3*n - 1)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (5*n + 2 + (2*n^2 + n + 10)*x + (5*n + 1)*x^2 + 2*x^3)*P[n - 1, x] + (n*x - (n - 2)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 16 + 20*x + 4*x^2; v[2] = 60 + 156*x + 76*x^2 + 8*x^3; for(n = 3, N, my(i = n); v[i] = (-(6*n + 6 + (4*n^2 + 2*n + 10)*x + (n^3 + n^2 + 9*n - 3)*x^2 + (3*n^2 + 2*n + 3)*x^3 + (2*n + 2)*x^4)*v[i - 2] - ((2*n + 2)*x + (n^2 - n + 2)*x^2 - (n^2 - 2*n + 3)*x^3 - (3*n - 1)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (5*n + 2 + (2*n^2 + n + 10)*x + (5*n + 1)*x^2 + 2*x^3)*v[i - 1] + (n*x - (n - 2)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 2*x)); 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.?
  - 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) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

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.