A200536 → basis A013609

OEIS matches ?

• A009972: 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,        14,          8;
  2:       36,       168,        292,         224,          64;
  3:      216,      1512,       4392,        6776,        5856,         2688,          512;
  4:     1296,     12096,      49248,      114240,      165136,       152320,        87552,        28672,          4096;
  5:     7776,     90720,     475200,     1471680,     2984160,      4139744,      3978880,      2616320,       1126400,        286720,         32768;
  6:    46656,    653184,    4183488,    16208640,    42310080,     78390144,    105702976,    104520192,      75217920,      38420480,      13221888,       2752512,        262144;
  7:   279936,   4572288,   34618752,   161046144,   514249344,   1192351104,   2070160512,   2734083968,    2760214016,    2119735296,    1218961408,     508985344,     145883136,     25690112,      2097152;
  8:  1679616,  31352832,  273964032,  1487517696,  5617092096,  15641966592,  33227873280,  54925559808,   71399743744,   73234079744,   59071774720,   37077254144,   17752784896,   6268387328,   1539309568,   234881024,    16777216;
  9: 10077696, 211631616, 2096160768, 13011425280, 56719512576, 184391230464, 463291508736, 919879317504, 1463179286016, 1878405369344, 1950905714688, 1635341008896, 1098172465152, 582767345664, 239015559168, 73106718720, 15703474176, 2113929216, 134217728;

The row polynomials satisfy P(n,x) = (6+14*x+8*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 + 14*x + 8*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 + 14*x + 8*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-185113cbc4b6c77b
• 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-185113cbc4b6c77b
• 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-185113cbc4b6c77b

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 A013609 e-185113cbc4b6c77b

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-982ad459e6d62a8e root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.