A100861 → reverse → x ↦ x+1

OEIS matches ?

• A000898: 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,     1;
  2:    2,     3,     1;
  3:    4,     9,     6,     1;
  4:   10,    28,    27,    10,     1;
  5:   26,    90,   115,    65,    15,     1;
  6:   76,   306,   480,   365,   135,    21,    1;
  7:  232,  1078,  2016,  1925,   980,   252,   28,   1;
  8:  764,  3984,  8596,  9856,  6405,  2324,  434,  36,  1;
  9: 2620, 15228, 37332, 49980, 39501, 18585, 4998, 702, 45, 1;

The row polynomials satisfy P(n,x) = (n-1+(n-1)*x)*P(n-2,x) + (1+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(n - 1 + (n - 1)*x)*P[n - 2, x] + (1 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + (n - 1)*x)*v[i - 2] + (1 + x)*v[i - 1]); 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-d0b0c6817e15c3a8
• 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-64733a2a119dbb75, e-d0b0c6817e15c3a8
• 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-64733a2a119dbb75, e-d0b0c6817e15c3a8
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-64733a2a119dbb75

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-d0b0c6817e15c3a8

  Geometric basis map (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
• definition_equality e-4ca095e71a768574 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-64733a2a119dbb75

  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 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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