A188440 → basis A007318

OEIS matches ?

• A056451: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A074872: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A162962: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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;
  2:  3,   2;
  3:  3,   2;
  4:  9,  12,   4;
  5:  9,  12,   4;
  6: 27,  54,  36,  8;
  7: 27,  54,  36,  8;
  8: 81, 216, 216, 96, 16;
  9: 81, 216, 216, 96, 16;

The row polynomials satisfy P(n,x) = (2*n-1+(2*n-2)*x)*P(n-1,x) - (6+10*x+4*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[(2*n - 1 + (2*n - 2)*x)*P[n - 1, x] - (6 + 10*x + 4*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] = (2*n - 1 + (2*n - 2)*x)*v[i - 1] - (6 + 10*x + 4*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-c5ecb99958d189a5
• 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-0b9b63070f7f0adc, e-c5ecb99958d189a5
• 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-0b9b63070f7f0adc, e-c5ecb99958d189a5
• 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-0b9b63070f7f0adc

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-0b9b63070f7f0adc

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-b186cc46021ee4e1 root: theorem check

      Parent-row provers of polytool polytool ?

• definition_equality e-e785025134c5ea24 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-c5ecb99958d189a5

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ parent_provers periodic_product e-b186cc46021ee4e1 (expanded above)

Related sequences ?

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

• A152198 → basis A013610
• A274498 → x ↦ x+1
• A274496 → basis A013610