A133087 → θ+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: 2,  2;
  2: 1,  4,   3;
  3: 2, 10,  12,   4;
  4: 1,  8,  18,  16,   5;
  5: 2, 18,  48,  56,  30,    6;
  6: 1, 12,  45,  80,  75,   36,   7;
  7: 2, 26, 108, 220, 250,  162,  56,   8;
  8: 1, 16,  84, 224, 350,  336, 196,  64,  9;
  9: 2, 34, 192, 560, 980, 1092, 784, 352, 90, 10;

The row polynomials satisfy (2+2*n*x+(n^2-n)*x^2)*P(n,x) = -(6*n-10+(9*n^2-11*n-6)*x+(3*n^3-7*n)*x^2+(2*n^3-3*n^2-n+2)*x^3)*P(n-1,x) + (3+(6*n+4)*x+(3*n^2+9*n)*x^2+(5*n^2+3*n-2)*x^3+(2*n^2-1)*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[(-(6*n - 10 + (9*n^2 - 11*n - 6)*x + (3*n^3 - 7*n)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*P[n - 1, x] + (3 + (6*n + 4)*x + (3*n^2 + 9*n)*x^2 + (5*n^2 + 3*n - 2)*x^3 + (2*n^2 - 1)*x^4)*D[P[n - 1, x], x])/(2 + 2*n*x + (n^2 - n)*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] = (-(6*n - 10 + (9*n^2 - 11*n - 6)*x + (3*n^3 - 7*n)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*v[i - 1] + (3 + (6*n + 4)*x + (3*n^2 + 9*n)*x^2 + (5*n^2 + 3*n - 2)*x^3 + (2*n^2 - 1)*x^4)*deriv(v[i - 1], x))/(2 + 2*n*x + (n^2 - n)*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-b7809f2be8630f6c
• 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-b7809f2be8630f6c
• 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-b7809f2be8630f6c

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 euler_polynomial, shift 0 e-b7809f2be8630f6c

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.