A100861 → reverse → x ↦ x/(x+1)

OEIS matches ?

• A047974: 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: 0, 1;
  2: 0, 1, 2;
  3: 0, 0, 3,  4;
  4: 0, 0, 3, 12,  10;
  5: 0, 0, 0, 15,  40,  26;
  6: 0, 0, 0, 15,  90, 150,   76;
  7: 0, 0, 0,  0, 105, 420,  546,  232;
  8: 0, 0, 0,  0, 105, 840, 2100, 2128,  764;
  9: 0, 0, 0,  0,   0, 945, 5040, 9828, 8352, 2620;

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

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-fc862e6fd76dbdc8
• 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-fc862e6fd76dbdc8
• 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-fc862e6fd76dbdc8

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 mobius, shift 0 e-fc862e6fd76dbdc8

  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

          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]
            no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

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

• A100861 → basis A007318

References

1. [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