A272866 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 16\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	5	5													11
2	1	10	35	50	25											121
3	1	15	90	275	450	375	125									1331
4	1	20	170	800	2275	4000	4250	2500	625							14641
5	1	25	275	1750	7125	19375	35625	43750	34375	15625	3125					161051
6	1	30	405	3250	17250	63750	168125	318750	431250	406250	253125	93750	15625			1771561
7	1	35	560	5425	35525	166250	573125	1478125	2865625	4156250	4440625	3390625	1750000	546875	78125	19487171

Show all 15 stored rows

Generated b-files: A001020 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-061162b1b3be9d7f · revision 1 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-d62194962de0e59f

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A001020: row sums, 15 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,  5,   5;
  2: 1, 10,  35,    50,     25;
  3: 1, 15,  90,   275,    450,    375,     125;
  4: 1, 20, 170,   800,   2275,   4000,    4250,     2500,      625;
  5: 1, 25, 275,  1750,   7125,  19375,   35625,    43750,    34375,     15625,      3125;
  6: 1, 30, 405,  3250,  17250,  63750,  168125,   318750,   431250,    406250,    253125,     93750,     15625;
  7: 1, 35, 560,  5425,  35525, 166250,  573125,  1478125,  2865625,   4156250,   4440625,   3390625,   1750000,    546875,     78125;
  8: 1, 40, 740,  8400,  65450, 371000, 1582000,  5175000, 13121875,  25875000,  39550000,  46375000,  40906250,  26250000,  11562500,   3125000,   390625;
  9: 1, 45, 945, 12300, 111150, 740250, 3764250, 14940000, 46906875, 117359375, 234534375, 373500000, 470531250, 462656250, 347343750, 192187500, 73828125, 17578125, 1953125;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+7*x+16*x^2+15*x^3+5*x^4)*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[((1 + 7*x + 16*x^2 + 15*x^3 + 5*x^4)*P[n - 1, x])/(1 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 7*x + 16*x^2 + 15*x^3 + 5*x^4)*v[i - 1])/(1 + 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 [-1, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-06d8b869d62a8f95
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-06d8b869d62a8f95
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-06d8b869d62a8f95

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 mobius, shift 0 e-06d8b869d62a8f95

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - preserver_image reversal, shift 0 e-0e058c428dfdfeb6
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check
    
    Parent-row provers of polytool polytool ?
  - ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)

Related sequences ?

Same fingerprint (1)

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

A272866 → basis A007318