Real-rooted atlas

A159854 → x ↦ x/(x+1)

nonnegative coefficients only rational roots

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

Recurrence
\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 3\,x + 2\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
00011
100123
2001449
3001612827
4001824321681
50011040808032243
6001126016024019264729
700114842805606724481282187
80011611244811201792179210242566561
9001181446722016403253764608230451219683
1000120180960336080641344015360115205120102459049
1100122220132052801478429568422404224028160112642048177147
120012426417607920253445913610137612672011264067584245764096531441
1300126312228811440411841098242196483294723660802928641597445324881921594323
14001283642912160166406419219243929676876810250241025024745472372736114688163844782969

Generated b-files: A000244 (row sums), A013609, A038207 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-890099793b4c816b · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-11aec07ce60b6d61

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): yes

OEIS matches ?

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: 0, 0, 1;
  1: 0, 0, 1,  2;
  2: 0, 0, 1,  4,   4;
  3: 0, 0, 1,  6,  12,   8;
  4: 0, 0, 1,  8,  24,  32,   16;
  5: 0, 0, 1, 10,  40,  80,   80,   32;
  6: 0, 0, 1, 12,  60, 160,  240,  192,   64;
  7: 0, 0, 1, 14,  84, 280,  560,  672,  448,  128;
  8: 0, 0, 1, 16, 112, 448, 1120, 1792, 1792, 1024,  256;
  9: 0, 0, 1, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1+3*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = x^2.
%t Mathematica
P[0, x_] := x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 3*x + 2*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 3*x + 2*x^2)*v[i - 1])/(1 + x)); 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.

A013609: triangle, offset 0, from the row P0, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  4,   4;
  3: 1,  6,  12,   8;
  4: 1,  8,  24,  32,   16;
  5: 1, 10,  40,  80,   80,   32;
  6: 1, 12,  60, 160,  240,  192,   64;
  7: 1, 14,  84, 280,  560,  672,  448,  128;
  8: 1, 16, 112, 448, 1120, 1792, 1792, 1024,  256;
  9: 1, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (1+x)*P(n,x) = (1+3*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = x^2.
%t Mathematica
P[0, x_] := x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 3*x + 2*x^2)*P[n - 1, x])/(1 + x)]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 3*x + 2*x^2)*v[i - 1])/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038207: reversed triangle, offset 0, from the row P0, low zeros dropped
%e Example
Triangle begins:
  0:   1;
  1:   2,    1;
  2:   4,    4,    1;
  3:   8,   12,    6,    1;
  4:  16,   32,   24,    8,    1;
  5:  32,   80,   80,   40,   10,    1;
  6:  64,  192,  240,  160,   60,   12,   1;
  7: 128,  448,  672,  560,  280,   84,  14,   1;
  8: 256, 1024, 1792, 1792, 1120,  448, 112,  16,  1;
  9: 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (1+x)*P(n,x) = (1+3*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = x^2.
%t Mathematica
P[0, x_] := x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 3*x + 2*x^2)*P[n - 1, x])/(1 + x)]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 3*x + 2*x^2)*v[i - 1])/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-dd3ac95d772228bd
  • 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-dd3ac95d772228bd
  • 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-dd3ac95d772228bd

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

Same fingerprint (67)

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