A130595: Triangle read by rows: lower triangular matrix which is inverse to Pascal's triangle (A007318) regarded as a lower triangular matrix.

OEIS matches ?

• A130595: triangle, 78 terms agree, 12 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 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:
   1:  1;
   2: -1,  1;
   3:  1, -2,   1;
   4: -1,  3,  -3,   1;
   5:  1, -4,   6,  -4,    1;
   6: -1,  5, -10,  10,   -5,   1;
   7:  1, -6,  15, -20,   15,  -6,   1;
   8: -1,  7, -21,  35,  -35,  21,  -7,  1;
   9:  1, -8,  28, -56,   70, -56,  28, -8,  1;
  10: -1,  9, -36,  84, -126, 126, -84, 36, -9, 1;

The row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(1 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
  0:  1;
  1: -1,  1;
  2:  1, -2,   1;
  3: -1,  3,  -3,   1;
  4:  1, -4,   6,  -4,    1;
  5: -1,  5, -10,  10,   -5,   1;
  6:  1, -6,  15, -20,   15,  -6,   1;
  7: -1,  7, -21,  35,  -35,  21,  -7,  1;
  8:  1, -8,  28, -56,   70, -56,  28, -8,  1;
  9: -1,  9, -36,  84, -126, 126, -84, 36, -9, 1;

The row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-26ea87317172461f

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 A130595 e-26ea87317172461f

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-6e85c0971e84e272 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-ad67990681a86223

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)

Related sequences ?

Same fingerprint (2)

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

• A021012: Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
• A023531 → x ↦ x−1