A100861 → reverse → basis A013610

OEIS matches ?

• A294119: 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,     3;
  2:    2,     9,      9;
  3:    4,    27,     54,      27;
  4:   10,    84,    243,     270,      81;
  5:   26,   270,   1035,    1755,    1215,     243;
  6:   76,   918,   4320,    9855,   10935,    5103,     729;
  7:  232,  3234,  18144,   51975,   79380,   61236,   20412,    2187;
  8:  764, 11952,  77364,  266112,  518805,  564732,  316386,   78732,   6561;
  9: 2620, 45684, 335988, 1349460, 3199581, 4516155, 3643542, 1535274, 295245, 19683;

The row polynomials satisfy P(n,x) = (n-1+(3*n-3)*x)*P(n-2,x) + (1+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.

P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(n - 1 + (3*n - 3)*x)*P[n - 2, x] + (1 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + (3*n - 3)*x)*v[i - 2] + (1 + 3*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-a15a4571c9a83032
• 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-a15a4571c9a83032
• 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-a15a4571c9a83032

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

• basis_map A013610 e-a15a4571c9a83032

  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

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