A133087 → reverse

OEIS matches ?

• A133087: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A084221: 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,  2;
  2: 1,  2,  1;
  3: 1,  4,  5,   2;
  4: 1,  4,  6,   4,   1;
  5: 1,  6, 14,  16,   9,   2;
  6: 1,  6, 15,  20,  15,   6,   1;
  7: 1,  8, 27,  50,  55,  36,  13,  2;
  8: 1,  8, 28,  56,  70,  56,  28,  8,  1;
  9: 1, 10, 44, 112, 182, 196, 140, 64, 17, 2;

The row polynomials satisfy P(n,x) = (1+2*n*x+(3*n-3)*x^2)*P(n-1,x) - (2*x+5*x^2+3*x^3)*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[(1 + 2*n*x + (3*n - 3)*x^2)*P[n - 1, x] - (2*x + 5*x^2 + 3*x^3)*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] = (1 + 2*n*x + (3*n - 3)*x^2)*v[i - 1] - (2*x + 5*x^2 + 3*x^3)*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)

Triangle begins:
  0: 1;
  1: 2,  1;
  2: 1,  2,  1;
  3: 2,  5,  4,   1;
  4: 1,  4,  6,   4,   1;
  5: 2,  9, 16,  14,   6,   1;
  6: 1,  6, 15,  20,  15,   6,   1;
  7: 2, 13, 36,  55,  50,  27,   8,  1;
  8: 1,  8, 28,  56,  70,  56,  28,  8,  1;
  9: 2, 17, 64, 140, 196, 182, 112, 44, 10, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*n*x+(3*n-3)*x^2)*P(n-1,x) - (2*x+5*x^2+3*x^3)*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[(1 + 2*n*x + (3*n - 3)*x^2)*P[n - 1, x] - (2*x + 5*x^2 + 3*x^3)*D[P[n - 1, x], x]];
Table[Reverse[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 + 2*n*x + (3*n - 3)*x^2)*v[i - 1] - (2*x + 5*x^2 + 3*x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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 >= 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-7d9675c994dd8650
• 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-7d9675c994dd8650
• 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-7d9675c994dd8650

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

• preserver_image reversal, shift 0 e-7d9675c994dd8650

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): A133080 * A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A007318 e-3537980ff71b1a89

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-4d02ac33f5b2fbc9 root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-aa3a6b002de14e0c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image affine, shift 0 e-28a1cf3333f63fa4

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-4d02ac33f5b2fbc9 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A133087: A133080 * A007318.