A272867 → basis A013611

OEIS matches ?

• A009990: 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,        24,         16;
  2:       36,       288,        768,         768,          256;
  3:      216,      2592,      12096,       27648,        32256,         18432,          4096;
  4:     1296,     20736,     138240,      497664,      1050624,       1327104,        983040,         393216,          65536;
  5:     7776,    155520,    1347840,     6635520,     20459520,      41140224,      54558720,       47185920,       25559040,         7864320,         1048576;
  6:    46656,   1119744,   11943936,    74649600,    303575040,     844038144,    1642070016,     2250768384,     2158755840,      1415577600,       603979776,       150994944,        16777216;
  7:   279936,   7838208,   99283968,   752467968,   3804143616,   13544423424,   34966536192,    66418900992,    93244096512,     96315899904,     72137834496,     38050725888,     13388218368,      2818572288,      268435456;
  8:  1679616,  53747712,  788299776,  7023034368,  42472636416,  184605474816,  595731677184,  1454421049344,  2712982781952,   3878456131584,   4236314148864,   3500666781696,   2147752083456,    947040288768,   283467841536,    51539607552,    4294967296;
  9: 10077696, 362797056, 6046617600, 61917364224, 436001439744, 2239344672768, 8684483641344, 25977774145536, 60715708710912, 111653060345856, 161908556562432, 184730838368256, 164683541643264, 113238960242688, 58793807314944, 22265110462464, 5798205849600, 927712935936, 68719476736;

The row polynomials satisfy P(n,x) = (6+24*x+16*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 + 24*x + 16*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 + 24*x + 16*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-bced44278fba7ed1
• 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-bced44278fba7ed1
• 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-bced44278fba7ed1

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 A013611 e-bced44278fba7ed1

  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.