A118931: Triangle, read by rows, where T(n,k) = n!/(k!(n-3k)!3^k) for n>=3k>=0.

A118931 nonnegative coefficients

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

Recurrence from OEIS A118931

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

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

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	P_n(1)
0	1			1
1	1			1
2	1			1
3	1	2		3
4	1	8		9
5	1	20		21
6	1	40	40	81
7	1	70	280	351

Show all 15 stored rows

Generated b-files: A001470 (row sums), A118931 ?

Computed in your browser ?

Technical details

Node id: n-6f93222f142b1c00 · revision 1 · JSON
Definition source: oeis_catalog:A118931
Fingerprint: f6-96e89084f9f5d8f5

Observed on the stored rows ?

Degrees: 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4 (irregular)
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 ?

A001470: 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 ?
A118931: triangle, 45 terms agree, 15 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 1001 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;
  2: 1;
  3: 1,   2;
  4: 1,   8;
  5: 1,  20;
  6: 1,  40,   40;
  7: 1,  70,  280;
  8: 1, 112, 1120;
  9: 1, 168, 3360, 2240;

%F Formula

The row polynomials satisfy P(n,x) = P(n-1,x) + (n^2-3*n+2)*x*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + (n^2 - 3*n + 2)*x*P[n - 3, 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; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = v[i - 1] + (n^2 - 3*n + 2)*x*v[i - 3]); 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.

A118931: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1,   2;
  4: 1,   8;
  5: 1,  20;
  6: 1,  40,   40;
  7: 1,  70,  280;
  8: 1, 112, 1120;
  9: 1, 168, 3360, 2240;

%F Formula

The row polynomials satisfy P(n,x) = P(n-1,x) + (n^2-3*n+2)*x*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + (n^2 - 3*n + 2)*x*P[n - 3, 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; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = v[i - 1] + (n^2 - 3*n + 2)*x*v[i - 3]); 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 (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-340a6e5084723b41
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-340a6e5084723b41
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-340a6e5084723b41
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-340a6e5084723b41

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

parent_provers paper_theorem e-340a6e5084723b41 root: theorem check

Parent-row provers of polytool polytool ?

Used in 6 proof steps

e-0142626e8a0fd8cd basis_map (A013611) → basis_map_A013611 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3…
e-29d505b34d506bd7 basis_map (A007318) → basis_map_A007318 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3…
e-9ee18672ede43293 basis_map (A013609) → basis_map_A013609 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3…
e-c35f8cdc3744c126 basis_map (A008277) → basis_map_A008277 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3…
e-ed0a9060e2227d5c basis_map (A013610) → basis_map_A013610 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3…
e-97aa57c096c47b30 preserver_image (affine, shift 0) → basis_map_A007318 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3…

Related sequences ?

No candidate equalities or related sequences.