A257624 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 6\,x + 3\,n\right)\,P_{n}(x) = \left(40 - 295\,x + 352\,x^{2} - 135\,x^{3} + 18\,x^{4} - 120\,n + 195\,n x - 270\,n x^{2} + 162\,n x^{3} - 54\,n x^{4} - 90\,n^{2} x + 81\,n^{2} x^{2} - 81\,n^{2} x^{3} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(15\,x - 117\,x^{2} + 183\,x^{3} - 135\,x^{4} + 54\,x^{5} - 45\,n x + 90\,n x^{2} - 126\,n x^{3} + 81\,n x^{4} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-37 + 98\,x - 51\,x^{2} + 18\,x^{3} + 39\,n - 39\,n x + 45\,n x^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x + 30\,x^{2} - 18\,x^{3} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 10 + 5\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	10	5							15
2	105	130	25						260
3	1040	2745	1290	125					5200
4	9695	49330	47340	11570	625				118560
5	86310	780005	1393640	681150	98810	3125			3043040
6	743605	11221290	34513095	30742100	8886555	821730	15625		86944000
7	6261340	150728825	749582370	1128849275	587133880	109372095	6730090	78125	2738736000

All 15 rows

Technical details

Node id: n-733f1c4823c75c08 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8b7b317e1c28b836

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:        10,           5;
  2:       105,         130,           25;
  3:      1040,        2745,         1290,          125;
  4:      9695,       49330,        47340,        11570,           625;
  5:     86310,      780005,      1393640,       681150,         98810,         3125;
  6:    743605,    11221290,     34513095,     30742100,       8886555,       821730,        15625;
  7:   6261340,   150728825,    749582370,   1128849275,     587133880,    109372095,      6730090,       78125;
  8:  51887595,  1925926570,  14755917010,  35310934350,   30707345420,  10229522870,   1297693590,    54621970,    390625;
  9: 425257010, 23719871925, 269777936160, 976521847740, 1335717945540, 738878961870, 167581205520, 15034715100, 440882010, 1953125;

%F Formula

The row polynomials satisfy (3*n-4+6*x)*P(n,x) = -(120*n-40+(90*n^2-195*n+295)*x+(27*n^3-81*n^2+270*n-352)*x^2+(81*n^2-162*n+135)*x^3+(54*n-18)*x^4)*P(n-2,x) - ((45*n-15)*x+(27*n^2-90*n+117)*x^2-(27*n^2-126*n+183)*x^3-(81*n-135)*x^4-54*x^5)*d/dx P(n-2,x) + (39*n-37+(18*n^2-39*n+98)*x+(45*n-51)*x^2+18*x^3)*P(n-1,x) + ((9*n-12)*x-(9*n-30)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 5*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 10 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(120*n - 40 + (90*n^2 - 195*n + 295)*x + (27*n^3 - 81*n^2 + 270*n - 352)*x^2 + (81*n^2 - 162*n + 135)*x^3 + (54*n - 18)*x^4)*P[n - 2, x] - ((45*n - 15)*x + (27*n^2 - 90*n + 117)*x^2 - (27*n^2 - 126*n + 183)*x^3 - (81*n - 135)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (39*n - 37 + (18*n^2 - 39*n + 98)*x + (45*n - 51)*x^2 + 18*x^3)*P[n - 1, x] + ((9*n - 12)*x - (9*n - 30)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n - 4 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 10 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(120*n - 40 + (90*n^2 - 195*n + 295)*x + (27*n^3 - 81*n^2 + 270*n - 352)*x^2 + (81*n^2 - 162*n + 135)*x^3 + (54*n - 18)*x^4)*v[i - 2] - ((45*n - 15)*x + (27*n^2 - 90*n + 117)*x^2 - (27*n^2 - 126*n + 183)*x^3 - (81*n - 135)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (39*n - 37 + (18*n^2 - 39*n + 98)*x + (45*n - 51)*x^2 + 18*x^3)*v[i - 1] + ((9*n - 12)*x - (9*n - 30)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n - 4 + 6*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.?
    
    ↑ 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.