Real-rooted atlas

A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).

A135838 palindromic γ-positive nonnegative coefficients only rational roots

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

Recurrence from OEIS A135838
\(P_{n}(x) = \left(2 + 4\,x + 2\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
22428
341212432
44162416464
58408080408256
6848120160120488512
716112336560560336112162048
8161284488961120896448128164096
9322881152268840324032268811522883216384
103232014403840672080646720384014403203232768
11647043520105602112029568295682112010560352070464131072
1264768422414080316805068859136506883168014080422476864262144
131281664998436608915201647362196482196481647369152036608998416641281048576
1412817921164846592128128256256384384439296384384256256128128465921164817921282097152

Generated b-files: A094015 (row sums), A135838, A135838 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-a624e5aba8e1c39e · revision 2 · JSON
Definition source
oeis_catalog:A135838
Fingerprint
f6-77a7ed3b1d5f257e

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 0)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • 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:  1;
  1:  2,   2;
  2:  2,   4,    2;
  3:  4,  12,   12,    4;
  4:  4,  16,   24,   16,    4;
  5:  8,  40,   80,   80,   40,    8;
  6:  8,  48,  120,  160,  120,   48,    8;
  7: 16, 112,  336,  560,  560,  336,  112,   16;
  8: 16, 128,  448,  896, 1120,  896,  448,  128,  16;
  9: 32, 288, 1152, 2688, 4032, 4032, 2688, 1152, 288, 32;
%F Formula
The row polynomials satisfy P(n,x) = (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x + 2*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 4*x + 2*x^2)*v[i - 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.

A135838: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2,   2;
   3:  2,   4,    2;
   4:  4,  12,   12,    4;
   5:  4,  16,   24,   16,    4;
   6:  8,  40,   80,   80,   40,    8;
   7:  8,  48,  120,  160,  120,   48,    8;
   8: 16, 112,  336,  560,  560,  336,  112,   16;
   9: 16, 128,  448,  896, 1120,  896,  448,  128,  16;
  10: 32, 288, 1152, 2688, 4032, 4032, 2688, 1152, 288, 32;
%F Formula
The row polynomials satisfy P(n,x) = (2+4*x+2*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x + 2*x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = (2 + 4*x + 2*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A135838: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:  1;
   2:  2,   2;
   3:  2,   4,    2;
   4:  4,  12,   12,    4;
   5:  4,  16,   24,   16,    4;
   6:  8,  40,   80,   80,   40,    8;
   7:  8,  48,  120,  160,  120,   48,    8;
   8: 16, 112,  336,  560,  560,  336,  112,   16;
   9: 16, 128,  448,  896, 1120,  896,  448,  128,  16;
  10: 32, 288, 1152, 2688, 4032, 4032, 2688, 1152, 288, 32;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+4*x+2*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x + 2*x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = (2 + 4*x + 2*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); 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 >= 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-07f7d31230240b98, e-51cb02553cf4a8ac
  • 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-07f7d31230240b98, e-51cb02553cf4a8ac
  • 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-07f7d31230240b98, e-51cb02553cf4a8ac

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 15 proof steps

Same fingerprint (59)

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