A249250 → D+1

nonnegative coefficients

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

Recurrence

\(\left(13\,x - 3\,x^{2} - 9\,x^{3} - x^{4} + 2\,n + n x - 22\,n x^{2} + 6\,n x^{3} + n x^{4} - n^{2} - 9\,n^{2} x + 9\,n^{2} x^{2} + n^{2} x^{3} - 3\,n^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(13\,x - 3\,x^{2} - 9\,x^{3} - x^{4} - n + 28\,n x + 8\,n x^{2} - 42\,n x^{3} - 5\,n x^{4} - 4\,n^{2} - 45\,n^{2} x^{2} - 5\,n^{2} x^{3} - 3\,n^{3} - 15\,n^{3} x\right)\,P_{n-1}(x) + \left(x + 16\,x^{2} + 71\,x^{3} + 107\,x^{4} + 20\,x^{5} + x^{6} + 4\,n x + 50\,n x^{2} + 105\,n x^{3} + 20\,n x^{4} + n x^{5} + 3\,n^{2} x + 30\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	3							7
2	7	12	3						22
3	10	39	24	3					76
4	13	84	108	36	3				244
5	16	147	336	222	48	3			772
6	19	228	789	912	381	60	3		2392
7	22	327	1548	2781	1974	585	72	3	7312

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-19386060e5ce678c · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4e27f9bcc8a4ab34

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1:  4,   3;
  2:  7,  12,    3;
  3: 10,  39,   24,     3;
  4: 13,  84,  108,    36,     3;
  5: 16, 147,  336,   222,    48,     3;
  6: 19, 228,  789,   912,   381,    60,    3;
  7: 22, 327, 1548,  2781,  1974,   585,   72,    3;
  8: 25, 444, 2694,  6828,  7500,  3684,  834,   84,  3;
  9: 28, 579, 4308, 14376, 22548, 16890, 6204, 1128, 96, 3;

%F Formula

The row polynomials satisfy -(3*n^3+n^2-2*n-(3*n^3-9*n^2+n+13)*x-(9*n^2-22*n-3)*x^2-(n^2+6*n-9)*x^3-(n-1)*x^4)*P(n,x) = -(3*n^3+4*n^2+n+(15*n^3-28*n-13)*x+(45*n^2-8*n+3)*x^2+(5*n^2+42*n+9)*x^3+(5*n+1)*x^4)*P(n-1,x) + ((3*n^2+4*n+1)*x+(30*n^2+50*n+16)*x^2+(3*n^2+105*n+71)*x^3+(20*n+107)*x^4+(n+20)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n^3 + 4*n^2 + n + (15*n^3 - 28*n - 13)*x + (45*n^2 - 8*n + 3)*x^2 + (5*n^2 + 42*n + 9)*x^3 + (5*n + 1)*x^4)*P[n - 1, x] + ((3*n^2 + 4*n + 1)*x + (30*n^2 + 50*n + 16)*x^2 + (3*n^2 + 105*n + 71)*x^3 + (20*n + 107)*x^4 + (n + 20)*x^5 + x^6)*D[P[n - 1, x], x])/(-3*n^3 - n^2 + 2*n + (3*n^3 - 9*n^2 + n + 13)*x + (9*n^2 - 22*n - 3)*x^2 + (n^2 + 6*n - 9)*x^3 + (n - 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(3*n^3 + 4*n^2 + n + (15*n^3 - 28*n - 13)*x + (45*n^2 - 8*n + 3)*x^2 + (5*n^2 + 42*n + 9)*x^3 + (5*n + 1)*x^4)*v[i - 1] + ((3*n^2 + 4*n + 1)*x + (30*n^2 + 50*n + 16)*x^2 + (3*n^2 + 105*n + 71)*x^3 + (20*n + 107)*x^4 + (n + 20)*x^5 + x^6)*deriv(v[i - 1], x))/(-3*n^3 - n^2 + 2*n + (3*n^3 - 9*n^2 + n + 13)*x + (9*n^2 - 22*n - 3)*x^2 + (n^2 + 6*n - 9)*x^3 + (n - 1)*x^4)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-747d16c15ece4cbb
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-747d16c15ece4cbb
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-747d16c15ece4cbb
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-747d16c15ece4cbb

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image derivative_polynomial, shift 0 e-747d16c15ece4cbb

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-86238c6a50aeb955 root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

No candidate equalities or related sequences.

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.