A122542 → 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

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

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

\(P_{2}(x) = 1 + x\)

\(P_{3}(x) = 2 + 4\,x + x^{2}\)

\(P_{4}(x) = 2 + 10\,x + 7\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	1														2
3	2	4	1													7
4	2	10	7	1												20
5	2	18	26	10	1											57
6	2	26	66	50	13	1										158
7	2	34	130	166	82	16	1									431
8	2	42	218	418	338	122	19	1								1160
9	2	50	330	870	1042	602	170	22	1							3089
10	2	58	466	1586	2610	2202	978	226	25	1						8154
11	2	66	626	2630	5618	6474	4146	1486	290	28	1					21367
12	2	74	810	4066	10802	16154	14026	7170	2146	362	31	1				55644
13	2	82	1018	5958	19058	35546	39930	27470	11618	2978	442	34	1			144137
14	2	90	1250	8370	31442	70906	99330	88034	49762	17882	4002	530	37	1		371638
15	2	98	1506	11366	49170	130826	221954	244878	177506	84722	26402	5238	626	40	1	954335

Generated b-files: A292400 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-6115e40a0e752de9 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-bf3a8403ddb100bf

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 -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

OEIS matches ?

A292400: row sums, 14 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: 1;
   2: 1,  1;
   3: 2,  4,   1;
   4: 2, 10,   7,    1;
   5: 2, 18,  26,   10,    1;
   6: 2, 26,  66,   50,   13,    1;
   7: 2, 34, 130,  166,   82,   16,   1;
   8: 2, 42, 218,  418,  338,  122,  19,   1;
   9: 2, 50, 330,  870, 1042,  602, 170,  22,  1;
  10: 2, 58, 466, 1586, 2610, 2202, 978, 226, 25, 1;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*x+2*x^2)*P(n-3,x) - (1+x^2)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 2 + 4*x + x^2, P(4,x) = 2 + 10*x + 7*x^2 + x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 2 + 4*x + x^2;
P[4, x_] := 2 + 10*x + 7*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*x + 2*x^2)*P[n - 3, x] - (1 + x^2)*P[n - 2, x] + (2 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 2 + 4*x + x^2; v[4] = 2 + 10*x + 7*x^2 + x^3; for(n = 5, N, my(i = n); v[i] = -x^2*v[i - 4] - (2*x + 2*x^2)*v[i - 3] - (1 + x^2)*v[i - 2] + (2 + 2*x)*v[i - 1]); 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-c74a6074373531d8
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-c74a6074373531d8
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-c74a6074373531d8
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-c74a6074373531d8

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-c74a6074373531d8

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-e90fe0b3eba80839 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.