A166344 → D+1

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 7 + 8\,x + x^{2}\)

\(P_{2}(x) = 16 + 45\,x + 18\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	7	8	1								16
2	16	45	18	1							80
3	35	214	192	38	1						480
4	74	885	1624	698	78	1					3360
5	153	3318	11327	9580	2343	158	1				26880
6	312	11633	68382	104535	49212	7527	318	1			241920
7	631	38950	371942	953874	797040	232554	23570	638	1		2419200
8	1270	126309	1876008	7608152	10505484	5377890	1042136	72672	1278	1	26611200

All 15 rows

Technical details

Node id: n-63d34f3b7f5dc912 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-1216106ea1ae9807

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:    7,       8,        1;
   2:   16,      45,       18,         1;
   3:   35,     214,      192,        38,          1;
   4:   74,     885,     1624,       698,         78,          1;
   5:  153,    3318,    11327,      9580,       2343,        158,         1;
   6:  312,   11633,    68382,    104535,      49212,       7527,       318,         1;
   7:  631,   38950,   371942,    953874,     797040,     232554,     23570,       638,        1;
   8: 1270,  126309,  1876008,   7608152,   10505484,    5377890,   1042136,     72672,     1278,      1;
   9: 2549,  400510,  8953413,  54862312,  118040882,   98927220,  33413050,   4509992,   221913,   2558,    1;
  10: 5108, 1249953, 41012638, 366564853, 1173115872, 1518776714, 834895636, 195982602, 19065212, 673493, 5118, 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) = 7 + 8*x + x^2, P(2,x) = 16 + 45*x + 18*x^2 + x^3.

%t Mathematica

P[1, x_] := 7 + 8*x + x^2;
P[2, x_] := 16 + 45*x + 18*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] = 7 + 8*x + x^2; v[2] = 16 + 45*x + 18*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
  - 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.