basis-map coefficients of A297191, shifted → basis A013610

OEIS matches ?

• A001025: 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,  6,    9;
  2: 1, 12,   54,   108,     81;
  3: 1, 18,  135,   540,   1215,    1458,      729;
  4: 1, 24,  252,  1512,   5670,   13608,    20412,    17496,      6561;
  5: 1, 30,  405,  3240,  17010,   61236,   153090,   262440,    295245,    196830,      59049;
  6: 1, 36,  594,  5940,  40095,  192456,   673596,  1732104,   3247695,   4330260,    3897234,    2125764,     531441;
  7: 1, 42,  819,  9828,  81081,  486486,  2189187,  7505784,  19702683,  39405366,   59108049,   64481508,   48361131,    22320522,     4782969;
  8: 1, 48, 1080, 15120, 147420, 1061424,  5837832, 25019280,  84440070, 225173520,  472864392,  773778096,  967222620,   892820880,   573956280,   229582512,   43046721;
  9: 1, 54, 1377, 22032, 247860, 2082024, 13533156, 69599088, 287096238, 956987460, 2583866142, 5637526128, 9865670724, 13660159464, 14635885140, 11708708112, 6586148313, 2324522934, 387420489;

The row polynomials satisfy P(n,x) = (1+6*x+9*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 + 6*x + 9*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 + 6*x + 9*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-4aebfd11077ddb65
• 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-4aebfd11077ddb65
• 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-4aebfd11077ddb65

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 A013610 e-4aebfd11077ddb65

  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.