A159854 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x + x^{2} + n x\right)\,P_{n}(x) = \left(2 + 5\,x + 4\,x^{2} + x^{3} + n x + n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\,x + x^{2}\)

Rows ?

n	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	2	1															3
1	2	4	1														7
2	2	7	6	1													16
3	2	10	15	8	1												36
4	2	13	28	26	10	1											80
5	2	16	45	60	40	12	1										176
6	2	19	66	115	110	57	14	1									384
7	2	22	91	196	245	182	77	16	1								832
8	2	25	120	308	476	462	280	100	18	1							1792
9	2	28	153	456	840	1008	798	408	126	20	1						3840
10	2	31	190	645	1380	1974	1932	1290	570	155	22	1					8192
11	2	34	231	880	2145	3564	4158	3432	1980	770	187	24	1				17408
12	2	37	276	1166	3190	6039	8184	8052	5742	2915	1012	222	26	1			36864
13	2	40	325	1508	4576	9724	15015	17160	14586	9152	4147	1300	260	28	1		77824
14	2	43	378	1911	6370	15015	26026	33891	33462	25025	14014	5733	1638	301	30	1	163840

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Technical details

Node id: n-f8e9ef674215319f · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-83fe4810a7b0eff6

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

A045891: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 0, 2,  1;
  1: 0, 2,  4,   1;
  2: 0, 2,  7,   6,   1;
  3: 0, 2, 10,  15,   8,   1;
  4: 0, 2, 13,  28,  26,  10,    1;
  5: 0, 2, 16,  45,  60,  40,   12,   1;
  6: 0, 2, 19,  66, 115, 110,   57,  14,   1;
  7: 0, 2, 22,  91, 196, 245,  182,  77,  16,   1;
  8: 0, 2, 25, 120, 308, 476,  462, 280, 100,  18,  1;
  9: 0, 2, 28, 153, 456, 840, 1008, 798, 408, 126, 20, 1;

%F Formula

The row polynomials satisfy (2+(n+2)*x+x^2)*P(n,x) = (2+(n+5)*x+(n+4)*x^2+x^3)*P(n-1,x) for n >= 1, with P(0,x) = 2*x + x^2.

%t Mathematica

P[0, x_] := 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (n + 5)*x + (n + 4)*x^2 + x^3)*P[n - 1, x])/(2 + (n + 2)*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (n + 5)*x + (n + 4)*x^2 + x^3)*v[i - 1])/(2 + (n + 2)*x + x^2)); 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-44b6e709ec6246c9
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-44b6e709ec6246c9
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-44b6e709ec6246c9

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-44b6e709ec6246c9

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Riordan array (1-x,x/(1-x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-b62074ba45057e82 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.