A094638 → basis A008277

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,         1;
  2:       6,         9,          2;
  3:      24,        81,         47,           6;
  4:     120,       825,        935,         290,          24;
  5:     720,      9675,      19085,       10765,        2074,         120;
  6:    5040,    130095,     420665,      383635,      128884,       16884,        720;
  7:   40320,   1986705,   10165295,    14070070,     7325269,     1628200,     154188,      5040;
  8:  362880,  34096545,  270026015,   545634670,   412896589,   138383952,   21814316,   1561104,    40320;
  9: 3628800, 651100275, 7867597325, 22618716925, 23903955124, 11327186661, 2641571792, 310289436, 17356176, 362880;

The row polynomials satisfy P(n,x) = (n+1+n*x)*P(n-1,x) + n*x*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[(n + 1 + n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], 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 + n*x)*v[i - 1] + n*x*deriv(v[i - 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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-56658731411f5b0e
• 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-56658731411f5b0e
• 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-56658731411f5b0e

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 A008277 e-56658731411f5b0e

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-419a6747cb084155 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • preserver_image reversal, shift 0 e-986a227a8a459030

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in incre…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-5f21311bb67f8c1c root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • basis_map A007318 e-0b3ba9047fae59ed

          Geometric basis map (elementary) polytool ?

          • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

              Parent-row provers of polytool polytool ?

        • parent_provers root_domain e-2e5c10abc86c264d root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image reversal, shift 0 e-90e69a839bdbf74b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-419a6747cb084155 (expanded above)
            • parent_provers paper_theorem e-b29f35705824bef6 root: theorem check

              Parent-row provers of polytool polytool ?

        • preserver_image affine, shift 0 e-1ba7a8eebd893648

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers root_domain e-edbf92ee2e446027 (expanded above)
    • ↵ parent_provers paper_theorem e-b29f35705824bef6 (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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741