A100861 → 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: 1;
  2: 1,  2;
  3: 1,  6;
  4: 1, 12,    9;
  5: 1, 20,   45;
  6: 1, 30,  135,   60;
  7: 1, 42,  315,  420;
  8: 1, 56,  630, 1680,  525;
  9: 1, 72, 1134, 5040, 4725;

The row polynomials satisfy (2+(n^2+n)*x)*P(n,x) = (2+(n^2+5*n-4)*x+(n^3+2*n^2-n-2)*x^2)*P(n-1,x) - (4*x^2+(2*n^2+6*n+4)*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[Together[((2 + (n^2 + 5*n - 4)*x + (n^3 + 2*n^2 - n - 2)*x^2)*P[n - 1, x] - (4*x^2 + (2*n^2 + 6*n + 4)*x^3)*D[P[n - 1, x], x])/(2 + (n^2 + n)*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^2 + 5*n - 4)*x + (n^3 + 2*n^2 - n - 2)*x^2)*v[i - 1] - (4*x^2 + (2*n^2 + 6*n + 4)*x^3)*deriv(v[i - 1], x))/(2 + (n^2 + n)*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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-0e3cf0b8dd3f845e
• 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-0e3cf0b8dd3f845e
• 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-0e3cf0b8dd3f845e

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-0e3cf0b8dd3f845e

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • gamma_polynomial scale 1, shift 0 e-037c1a08736a9fd9

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang_derivative_lag e-e7942008ad3e1420 root: theorem check

          Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

        • preserver_image reversal, shift 0 e-7078061f6d32a158

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang_derivative_lag e-e7942008ad3e1420 (expanded above)
    • parent_provers paper_theorem e-8187ad69de0e901a root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-a3a653eabafbad6b

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Exponential Riordan array (1, x(1+x/2)).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • preserver_image reversal, shift 0 e-01d50252f4aaa594

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ gamma_polynomial scale 1, shift 0 e-037c1a08736a9fd9 (expanded above)
            • ↵ parent_provers paper_theorem e-8187ad69de0e901a (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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.