A249248 → 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

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

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

\(P_{1}(x) = 4 + 3\,x\)

\(P_{2}(x) = 8 + 13\,x + 3\,x^{2}\)

\(P_{3}(x) = 13 + 56\,x + 31\,x^{2} + 3\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	3							7
2	8	13	3						24
3	13	56	31	3					103
4	19	156	189	52	3				419
5	26	343	882	485	76	3			1815
6	34	653	2959	3211	1007	103	3		7970
7	43	1128	7938	16037	9168	1827	133	3	36277

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-d940fcd93338e90e · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c1cb0cbde74e47c7

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:  8,   13,     3;
  3: 13,   56,    31,      3;
  4: 19,  156,   189,     52,      3;
  5: 26,  343,   882,    485,     76,      3;
  6: 34,  653,  2959,   3211,   1007,    103,     3;
  7: 43, 1128,  7938,  16037,   9168,   1827,   133,    3;
  8: 53, 1816, 18276,  61018,  62735,  21978,  3026,  166,   3;
  9: 64, 2771, 37684, 189122, 331858, 197729, 46426, 4694, 202, 3;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = -(n^3+n^2-2*n)*x^2*P(n-4,x) - ((2*n^2+4*n)*x+(2*n^2+3*n)*x^2)*P(n-3,x) - (n+2-(2*n^2+2*n-2)*x+(n+1)*x^2)*P(n-2,x) + (2*n+3+(2*n+2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 3*x, P(2,x) = 8 + 13*x + 3*x^2, P(3,x) = 13 + 56*x + 31*x^2 + 3*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[2, x_] := 8 + 13*x + 3*x^2;
P[3, x_] := 13 + 56*x + 31*x^2 + 3*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + n^2 - 2*n)*x^2*P[n - 4, x] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*P[n - 3, x] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*P[n - 2, x] + (2*n + 3 + (2*n + 2)*x)*P[n - 1, x])/(n + 1)]];
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] = 4 + 3*x; v[3] = 8 + 13*x + 3*x^2; v[4] = 13 + 56*x + 31*x^2 + 3*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 + n^2 - 2*n)*x^2*v[i - 4] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*v[i - 3] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*v[i - 2] + (2*n + 3 + (2*n + 2)*x)*v[i - 1])/(n + 1)); 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-a92c2e36218285ab
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-a92c2e36218285ab
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-a92c2e36218285ab
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-a92c2e36218285ab

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-a92c2e36218285ab

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-2dcae0cd92b4b84a 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.