basis-map coefficients of A297191, shifted → basis A013611

OEIS matches ?

• A009969: 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: 1,  8,   16;
  2: 1, 16,   96,   256,    256;
  3: 1, 24,  240,  1280,   3840,    6144,     4096;
  4: 1, 32,  448,  3584,  17920,   57344,   114688,    131072,      65536;
  5: 1, 40,  720,  7680,  53760,  258048,   860160,   1966080,    2949120,     2621440,     1048576;
  6: 1, 48, 1056, 14080, 126720,  811008,  3784704,  12976128,   32440320,    57671680,    69206016,     50331648,     16777216;
  7: 1, 56, 1456, 23296, 256256, 2050048, 12300288,  56229888,  196804608,   524812288,  1049624576,   1526726656,   1526726656,    939524096,    268435456;
  8: 1, 64, 1920, 35840, 465920, 4472832, 32800768, 187432960,  843448320,  2998927360,  8396996608,  18320719872,  30534533120,  37580963840,  32212254720,  17179869184,   4294967296;
  9: 1, 72, 2448, 52224, 783360, 8773632, 76038144, 521404416, 2867724288, 12745441280, 45883588608, 133479530496, 311452237824, 574988746752, 821412495360, 876173328384, 657129996288, 309237645312, 68719476736;

The row polynomials satisfy P(n,x) = (1+8*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[(1 + 8*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] = (1 + 8*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-a80b60183a85d18e
• 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-a80b60183a85d18e
• 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-a80b60183a85d18e

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-a80b60183a85d18e

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)

Related sequences ?

No candidate equalities or related sequences.