Real-rooted atlas

A272867: Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.

A272867 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A272867
\(P_{n}(x) = \left(1 + 4\,x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11416
218188136
3112518851121216
41161003044543041001611296
51201657201770242417707201652017776
6124246140048151022413236102244815140024624146656
71283432408106613088458947733925894730884106612408343281279936

Show all 15 stored rows

Generated b-files: A000400 (row sums), A272867, A272867 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-d3c3090977105807 · revision 1 · JSON
Definition source
oeis_catalog:A272867
Fingerprint
f6-f3772222c7dff591

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: 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): no

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: 1,  4,   1;
  2: 1,  8,  18,    8,     1;
  3: 1, 12,  51,   88,    51,     12,      1;
  4: 1, 16, 100,  304,   454,    304,    100,      16,       1;
  5: 1, 20, 165,  720,  1770,   2424,   1770,     720,     165,      20,       1;
  6: 1, 24, 246, 1400,  4815,  10224,  13236,   10224,    4815,    1400,     246,      24,      1;
  7: 1, 28, 343, 2408, 10661,  30884,  58947,   73392,   58947,   30884,   10661,    2408,    343,     28,     1;
  8: 1, 32, 456, 3808, 20636,  75936, 193144,  340064,  411462,  340064,  193144,   75936,  20636,   3808,   456,   32,   1;
  9: 1, 36, 585, 5664, 36324, 162288, 517524, 1188576, 1964862, 2325976, 1964862, 1188576, 517524, 162288, 36324, 5664, 585, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*x+x^2)*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[(1 + 4*x + x^2)*P[n - 1, x]];
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 + 4*x + x^2)*v[i - 1]); 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.

A272867: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  4,   1;
  2: 1,  8,  18,    8,     1;
  3: 1, 12,  51,   88,    51,     12,      1;
  4: 1, 16, 100,  304,   454,    304,    100,      16,       1;
  5: 1, 20, 165,  720,  1770,   2424,   1770,     720,     165,      20,       1;
  6: 1, 24, 246, 1400,  4815,  10224,  13236,   10224,    4815,    1400,     246,      24,      1;
  7: 1, 28, 343, 2408, 10661,  30884,  58947,   73392,   58947,   30884,   10661,    2408,    343,     28,     1;
  8: 1, 32, 456, 3808, 20636,  75936, 193144,  340064,  411462,  340064,  193144,   75936,  20636,   3808,   456,   32,   1;
  9: 1, 36, 585, 5664, 36324, 162288, 517524, 1188576, 1964862, 2325976, 1964862, 1188576, 517524, 162288, 36324, 5664, 585, 36, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*x+x^2)*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[(1 + 4*x + x^2)*P[n - 1, x]];
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 + 4*x + x^2)*v[i - 1]); 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.

A272867: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  4,   1;
  2: 1,  8,  18,    8,     1;
  3: 1, 12,  51,   88,    51,     12,      1;
  4: 1, 16, 100,  304,   454,    304,    100,      16,       1;
  5: 1, 20, 165,  720,  1770,   2424,   1770,     720,     165,      20,       1;
  6: 1, 24, 246, 1400,  4815,  10224,  13236,   10224,    4815,    1400,     246,      24,      1;
  7: 1, 28, 343, 2408, 10661,  30884,  58947,   73392,   58947,   30884,   10661,    2408,    343,     28,     1;
  8: 1, 32, 456, 3808, 20636,  75936, 193144,  340064,  411462,  340064,  193144,   75936,  20636,   3808,   456,   32,   1;
  9: 1, 36, 585, 5664, 36324, 162288, 517524, 1188576, 1964862, 2325976, 1964862, 1188576, 517524, 162288, 36324, 5664, 585, 36, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+4*x+x^2)*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[(1 + 4*x + x^2)*P[n - 1, x]];
Table[Reverse[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 + 4*x + x^2)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); 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 (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-eab26daf7024ea74
  • 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-eab26daf7024ea74
  • 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-eab26daf7024ea74

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

Used in 6 proof steps

No candidate equalities or related sequences.