A103633 → basis A007318

OEIS matches ?

• A026549: 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,   1;
  2:  2,   3,   1;
  3:  2,   5,   4,   1;
  4:  4,  12,  13,   6,   1;
  5:  4,  16,  25,  19,   7,   1;
  6:  8,  36,  66,  63,  33,   9,   1;
  7:  8,  44, 102, 129,  96,  42,  10,  1;
  8: 16,  96, 248, 360, 321, 180,  62, 12,  1;
  9: 16, 112, 344, 608, 681, 501, 242, 74, 13, 1;

The row polynomials satisfy P(n,x) = -(3*n-4+(2*n-3)*x)*P(n-1,x) + (4+6*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(3*n - 4 + (2*n - 3)*x)*P[n - 1, x] + (4 + 6*x + 2*x^2)*D[P[n - 1, x], 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] = -(3*n - 4 + (2*n - 3)*x)*v[i - 1] + (4 + 6*x + 2*x^2)*deriv(v[i - 1], x)); 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, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-d5900f5204767b18
• 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-01d2a4c51ed4f80c, e-d5900f5204767b18
• 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-01d2a4c51ed4f80c, e-d5900f5204767b18
• 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-01d2a4c51ed4f80c

Proof chains

3 incoming proof steps. 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-01d2a4c51ed4f80c

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-35a33ce2a6b89415 root: theorem check

      Parent-row provers of polytool polytool ?

• definition_equality e-30145a7ad20344a7 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-d5900f5204767b18

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ parent_provers periodic_product e-35a33ce2a6b89415 (expanded above)

Related sequences ?

No candidate equalities or related sequences.