Real-rooted atlas

A026374: Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.

A026374 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 A026374
\(P_{n}(x) = \left(1 + 3\,x + x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
21315
3144110
416116125
51717177150
61930453091125
711039757539101250
81125814419514458121625
911370202339339202701311250
1011595330685873685330951513125
1111611042510151558155810154251101616250
121181416301770325839893258177063014118115625
1311915977124005028724772475028240077115919131250
1412119610713801919815533184831553391983801107119621178125

Generated b-files: A026374, A026374 (reversed triangle), A026383 (row sums), A162963 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-4e0b19699f4cf0db · revision 2 · JSON
Definition source
oeis_catalog:A026374
Fingerprint
f6-9c9bca530a2aafba

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): 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,  1;
  2: 1,  3,  1;
  3: 1,  4,  4,   1;
  4: 1,  6, 11,   6,   1;
  5: 1,  7, 17,  17,   7,   1;
  6: 1,  9, 30,  45,  30,   9,   1;
  7: 1, 10, 39,  75,  75,  39,  10,  1;
  8: 1, 12, 58, 144, 195, 144,  58, 12,  1;
  9: 1, 13, 70, 202, 339, 339, 202, 70, 13, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + 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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + 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.

A026374: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,  1;
  3: 1,  4,  4,   1;
  4: 1,  6, 11,   6,   1;
  5: 1,  7, 17,  17,   7,   1;
  6: 1,  9, 30,  45,  30,   9,   1;
  7: 1, 10, 39,  75,  75,  39,  10,  1;
  8: 1, 12, 58, 144, 195, 144,  58, 12,  1;
  9: 1, 13, 70, 202, 339, 339, 202, 70, 13, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); 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.

A026374: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,  1;
  3: 1,  4,  4,   1;
  4: 1,  6, 11,   6,   1;
  5: 1,  7, 17,  17,   7,   1;
  6: 1,  9, 30,  45,  30,   9,   1;
  7: 1, 10, 39,  75,  75,  39,  10,  1;
  8: 1, 12, 58, 144, 195, 144,  58, 12,  1;
  9: 1, 13, 70, 202, 339, 339, 202, 70, 13, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+3*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); 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 (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6e3ca670df3d0a74, e-b2e3690a28625635
  • 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-6e3ca670df3d0a74, e-b2e3690a28625635
  • 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-6e3ca670df3d0a74, e-b2e3690a28625635

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 14 proof steps

No candidate equalities or related sequences.