A133087 → D+1

OEIS matches ?

No OEIS matches.

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:  3,   1;
  2:  3,   4,   1;
  3:  7,  13,   7,   1;
  4:  5,  16,  18,   8,    1;
  5: 11,  41,  58,  38,   11,   1;
  6:  7,  36,  75,  80,   45,  12,   1;
  7: 15,  85, 201, 255,  185,  75,  15,   1;
  8:  9,  64, 196, 336,  350, 224,  84,  16,  1;
  9: 19, 145, 484, 924, 1106, 854, 420, 124, 19, 1;

The row polynomials satisfy (n^2-n-1+(2*n-2)*x+x^2)*P(n,x) = -(3*n^3-9*n^2+5*n+4+(2*n^3+n^2-13*n+11)*x+(4*n^2-5*n-2)*x^2+(2*n-3)*x^3)*P(n-1,x) + (3*n^2-n-1+(5*n^2+5*n-3)*x+(2*n^2+10*n+1)*x^2+(4*n+5)*x^3+2*x^4)*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[Together[(-(3*n^3 - 9*n^2 + 5*n + 4 + (2*n^3 + n^2 - 13*n + 11)*x + (4*n^2 - 5*n - 2)*x^2 + (2*n - 3)*x^3)*P[n - 1, x] + (3*n^2 - n - 1 + (5*n^2 + 5*n - 3)*x + (2*n^2 + 10*n + 1)*x^2 + (4*n + 5)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^2 - n - 1 + (2*n - 2)*x + x^2)]];
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^3 - 9*n^2 + 5*n + 4 + (2*n^3 + n^2 - 13*n + 11)*x + (4*n^2 - 5*n - 2)*x^2 + (2*n - 3)*x^3)*v[i - 1] + (3*n^2 - n - 1 + (5*n^2 + 5*n - 3)*x + (2*n^2 + 10*n + 1)*x^2 + (4*n + 5)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^2 - n - 1 + (2*n - 2)*x + x^2)); 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-d70a0ba3fa52a1a1
• 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-d70a0ba3fa52a1a1
• 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-d70a0ba3fa52a1a1

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 derivative_polynomial, shift 0 e-d70a0ba3fa52a1a1

  Laguerre’s theorem on differential operators [SP14] 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 ?

No candidate equalities or related sequences.

References

1. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.