A299989 → basis A007318

OEIS matches ?

No OEIS matches.

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:         8,         14,          7,          1;
  2:        64,        160,        148,         64,          13,           1;
  3:       512,       1664,       2208,       1560,         636,         150,          19,           1;
  4:      4096,      16384,      28160,      27392,       16656,        6576,        1688,         272,         25,          1;
  5:     32768,     155648,     327680,     404480,      325760,      179936,       69616,       18880,       3520,        430,        31,         1;
  6:    262144,    1441792,    3588096,    5357568,     5360640,     3798528,     1962304,      748672,     211056,      43440,      6348,       624,       37,        1;
  7:   2097152,   13107200,   37617664,   65830912,    78618624,    67909632,    43850240,    21561728,    8142784,    2362528,    522480,     86520,    10388,      854,      43,      1;
  8:  16777216,  117440512,  381681664,  765460480,  1061552128,  1080819712,   836878336,   503504896,  238362880,   89318656,  26497792,   6189568,  1124704,   155680,   15856,   1120,    49,    1;
  9: 134217728, 1040187392, 3774873600, 8531214336, 13466861568, 15781330944, 14241497088, 10130128896, 5764810752, 2648231424, 986257152, 297821952, 72632832, 14183232, 2185632, 259776, 22968, 1422, 55, 1;

The row polynomials satisfy P(n,x) = (8+6*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + x.

P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(8 + 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 + x; for(n = 1, N, my(i = n + 1); v[i] = (8 + 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-298e29fac11cdc8c
• 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-298e29fac11cdc8c
• 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-298e29fac11cdc8c

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-298e29fac11cdc8c

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) f…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-0e7ef206328a472c root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.