A152828: Triangle read by rows, A007318 rows repeated three times.

OEIS matches ?

• A152828: reversed 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 ?
• A173862: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A152828: 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.

Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1, 1;
  4: 1, 1;
  5: 1, 1;
  6: 1, 2, 1;
  7: 1, 2, 1;
  8: 1, 2, 1;
  9: 1, 3, 3, 1;

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

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 3, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

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] = (1 + 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)

Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1, 1;
  4: 1, 1;
  5: 1, 1;
  6: 1, 2, 1;
  7: 1, 2, 1;
  8: 1, 2, 1;
  9: 1, 3, 3, 1;

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

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 3, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

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] = (1 + 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)

Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1, 1;
  4: 1, 1;
  5: 1, 1;
  6: 1, 2, 1;
  7: 1, 2, 1;
  8: 1, 2, 1;
  9: 1, 3, 3, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 3, x]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 0, 9}] // Flatten

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] = (1 + x)*v[i - 3]); 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)

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-9670611e62c74d1a
• 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-9670611e62c74d1a
• 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-9670611e62c74d1a

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 periodic_product e-9670611e62c74d1a root: theorem check

  Parent-row provers of polytool polytool ?

Used in 8 proof steps

• e-131d94ab8eb56aaf basis_map (A007318) → shift_x_plus_one of Triangle read by rows, A007318 rows repeated three times.
• e-15d22f53a2fb1793 basis_map (A013611) → basis_map_A013611 of Triangle read by rows, A007318 rows repeated three times.
• e-1fe5b61acc156f66 basis_map (A013609) → basis_map_A013609 of Triangle read by rows, A007318 rows repeated three times.
• e-9d0df611e4535b39 basis_map (A013610) → basis_map_A013610 of Triangle read by rows, A007318 rows repeated three times.
• e-be32ce4d8d6a286f basis_map (A008277) → basis_map_A008277 of Triangle read by rows, A007318 rows repeated three times.
• e-345c88cc98fbdbfc preserver_image (affine, shift 0) → shift_x_plus_one of Triangle read by rows, A007318 rows repeated three times.
• e-809cfb1679ce14dc preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle read by rows, A007318 rows repeated three times.
• e-916a7875f7a82d69 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle read by rows, A007318 rows repeated three times.

Related sequences ?

Same fingerprint (2)

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

• A152828 → basis A013609
• A152828 → D