A272867 → basis A007318

OEIS matches ?

• A001022: 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 ?

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:        6,        6,         1;
  2:       36,       72,        48,        12,          1;
  3:      216,      648,       756,       432,        126,         18,          1;
  4:     1296,     5184,      8640,      7776,       4104,       1296,        240,         24,         1;
  5:     7776,    38880,     84240,    103680,      79920,      40176,      13320,       2880,       390,        30,         1;
  6:    46656,   279936,    746496,   1166400,    1185840,     824256,     400896,     137376,     32940,      5400,       576,       36,       1;
  7:   279936,  1959552,   6205248,  11757312,   14859936,   13226976,    8536752,    4053888,   1422792,    367416,     68796,     9072,     798,      42,      1;
  8:  1679616, 13436928,  49268736, 109734912,  165908736,  180278784,  145442304,   88770816,  41396832,  14795136,   4040064,   834624,  128016,   14112,   1056,    48,    1;
  9: 10077696, 90699264, 377913600, 967458816, 1703130624, 2186860032, 2120235264, 1585557504, 926448192, 425922624, 154408032, 44043264, 9815904, 1687392, 219024, 20736, 1350, 54, 1;

The row polynomials satisfy P(n,x) = (6+6*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(6 + 6*x + x^2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6 + 6*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)

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-da5b5b0d47f37943
• 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-da5b5b0d47f37943
• 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-da5b5b0d47f37943

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

• basis_map A007318 e-da5b5b0d47f37943

  Geometric basis map (elementary) polytool ?

  • premise (sequence): 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.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-eab26daf7024ea74 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.