A166343 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-4\,x + 6\,x^{2} - 2\,x^{3} - 2\,n + 3\,n x - 5\,n x^{2} + 4\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 2\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 4\,x^{3} - 4\,x^{4} + 2\,x^{5} - n x + 2\,n x^{2} - 4\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 7\,x - 4\,x^{2} + 2\,x^{3} + 3\,n - 3\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 3\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 13 + 14\,x + x^{2}\)

\(P_{2}(x) = 28 + 81\,x + 30\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	13	14	1								28
2	28	81	30	1							140
3	59	382	336	62	1						840
4	122	1557	2872	1202	126	1					5880
5	249	5766	19991	16780	3999	254	1				47040
6	504	20033	120078	183855	85596	12783	510	1			423360
7	1015	66646	649838	1676514	1396080	402570	39914	1022	1		4233600
8	2038	215157	3263784	13342784	18435564	9387042	1798328	122856	2046	1	46569600

All 15 rows

Technical details

Node id: n-f3d0389fa7364339 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-61e8cab8d65b78d2

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:   13,      14,        1;
   2:   28,      81,       30,         1;
   3:   59,     382,      336,        62,          1;
   4:  122,    1557,     2872,      1202,        126,          1;
   5:  249,    5766,    19991,     16780,       3999,        254,          1;
   6:  504,   20033,   120078,    183855,      85596,      12783,        510,         1;
   7: 1015,   66646,   649838,   1676514,    1396080,     402570,      39914,      1022,        1;
   8: 2038,  215157,  3263784,  13342784,   18435564,    9387042,    1798328,    122856,     2046,       1;
   9: 4085,  680110, 15524349,  95981416,  207085466,  173250084,   58164610,   7766216,   374769,    4094,    1;
  10: 8180, 2117937, 70926574, 639874669, 2055823776, 2662348850, 1459357732, 340470546, 32784476, 1136669, 8190, 1;

%F Formula

The row polynomials satisfy (n-1+2*x)*P(n,x) = -(2*n+(2*n^2-3*n+4)*x+(n^3-2*n^2+5*n-6)*x^2+(3*n^2-4*n+2)*x^3+2*n*x^4)*P(n-2,x) - (n*x+(n^2-2*n+2)*x^2-(n^2-4*n+4)*x^3-(3*n-4)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-2+(2*n^2-3*n+7)*x+(5*n-4)*x^2+2*x^3)*P(n-1,x) + ((n-1)*x-(n-3)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 13 + 14*x + x^2, P(2,x) = 28 + 81*x + 30*x^2 + x^3.

%t Mathematica

P[1, x_] := 13 + 14*x + x^2;
P[2, x_] := 28 + 81*x + 30*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*P[n - 2, x] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 13 + 14*x + x^2; v[2] = 28 + 81*x + 30*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + (2*n^2 - 3*n + 4)*x + (n^3 - 2*n^2 + 5*n - 6)*x^2 + (3*n^2 - 4*n + 2)*x^3 + 2*n*x^4)*v[i - 2] - (n*x + (n^2 - 2*n + 2)*x^2 - (n^2 - 4*n + 4)*x^3 - (3*n - 4)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 2 + (2*n^2 - 3*n + 7)*x + (5*n - 4)*x^2 + 2*x^3)*v[i - 1] + ((n - 1)*x - (n - 3)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 1 + 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) = 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.