A106800 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(2 - 16\,x + 24\,x^{2} - n + 12\,n x - 26\,n x^{2} - 2\,n^{2} x + 9\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 6\,x^{3} + n x^{2} - 5\,n x^{3} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-5 + 18\,x + 2\,n - 12\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x^{2} - n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 0\)

\(P_{3}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	P_n(1)
1							0
2							0
3	1						1
4	3	2					5
5	6	14	3				23
6	10	50	45	4			109
7	15	130	270	124	5		544
8	21	280	1050	1204	315	6	2876

Show all 15 stored rows

Generated b-files: A278677 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-bb621e5a7fb6b6af · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-f041ba2901346a21

Observed on the stored rows ?

Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
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 ?

A278677: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:  0;
   2:  0;
   3:  1;
   4:  3,   2;
   5:  6,  14,    3;
   6: 10,  50,   45,     4;
   7: 15, 130,  270,   124,     5;
   8: 21, 280, 1050,  1204,   315,     6;
   9: 28, 532, 3150,  6804,  4830,   762,    7;
  10: 36, 924, 7938, 27804, 38850, 18150, 1785, 8;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(n-2+(2*n^2-12*n+16)*x+(n^3-9*n^2+26*n-24)*x^2)*P(n-2,x) + ((n-2)*x^2+(n^2-5*n+6)*x^3)*d/dx P(n-2,x) + (2*n-5+(2*n^2-12*n+18)*x)*P(n-1,x) - (n-3)*x^2*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (2*n^2 - 12*n + 16)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*P[n - 2, x] + ((n - 2)*x^2 + (n^2 - 5*n + 6)*x^3)*D[P[n - 2, x], x] + (2*n - 5 + (2*n^2 - 12*n + 18)*x)*P[n - 1, x] - (n - 3)*x^2*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 4, N, my(i = n); v[i] = (-(n - 2 + (2*n^2 - 12*n + 16)*x + (n^3 - 9*n^2 + 26*n - 24)*x^2)*v[i - 2] + ((n - 2)*x^2 + (n^2 - 5*n + 6)*x^3)*deriv(v[i - 2], x) + (2*n - 5 + (2*n^2 - 12*n + 18)*x)*v[i - 1] - (n - 3)*x^2*deriv(v[i - 1], x))/(n - 3)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-4dce43a45fb5789a
Coefficients of one sign. Every nonzero row, n >= 1: 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-4dce43a45fb5789a
Hadamard kernel (Malo). Rows P_n, n >= 1, 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-4dce43a45fb5789a
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-4dce43a45fb5789a

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-4dce43a45fb5789a

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-2ab1dfe057e84c9d root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A008278 → D

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.