A139548 → basis A007318

OEIS matches ?

• A001027: 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:         8,          8,          2;
  2:        64,        128,         96,          32,           4;
  3:       512,       1536,       1920,        1280,         480,          96,           8;
  4:      4096,      16384,      28672,       28672,       17920,        7168,        1792,         256,          16;
  5:     32768,     163840,     368640,      491520,      430080,      258048,      107520,       30720,        5760,         640,         32;
  6:    262144,    1572864,    4325376,     7208960,     8110080,     6488064,     3784704,     1622016,      506880,      112640,      16896,       1536,        64;
  7:   2097152,   14680064,   47710208,    95420416,   131203072,   131203072,    98402304,    56229888,    24600576,     8200192,    2050048,     372736,     46592,      3584,      128;
  8:  16777216,  134217728,  503316480,  1174405120,  1908408320,  2290089984,  2099249152,  1499463680,   843448320,   374865920,  131203072,   35782656,   7454720,   1146880,   122880,    8192,    256;
  9: 134217728, 1207959552, 5133828096, 13690208256, 25669140480, 35936796672, 38931529728, 33369882624, 22941794304, 12745441280, 5735448576, 2085617664, 608305152, 140378112, 25067520, 3342336, 313344, 18432, 512;

The row polynomials satisfy P(n,x) = (8+8*x+2*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[(8 + 8*x + 2*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] = (8 + 8*x + 2*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-debf4dfb42326c05
• 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-debf4dfb42326c05
• 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-debf4dfb42326c05

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-debf4dfb42326c05

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 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-4755dd2fd7289161 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (5)

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

• A034870 → basis A013611
• basis-map coefficients of A297191, shifted → basis A013609
• A034870 → x ↦ x+1
• A139548 → basis A013611
• A034870 → x ↦ x/(x+1)