A196347 → basis A008277

OEIS matches ?

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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:         8,         10,           2;
  3:        48,        114,          54,           6;
  4:       384,       1560,        1320,         336,          24;
  5:      3840,      25320,       34200,       15000,        2400,         120;
  6:     46080,     478800,      972720,      655200,      176400,       19440,        720;
  7:    645120,   10377360,    30587760,    29741040,    11995200,     2187360,     176400,      5040;
  8:  10321920,  254217600,  1061827200,  1434343680,   813697920,   218453760,   28788480,   1774080,    40320;
  9: 185794560, 6956772480, 40513737600, 74101910400, 56848780800, 21085868160, 4038854400, 402796800, 19595520, 362880;

The row polynomials satisfy P(n,x) = (2*n+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[(2*n + 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] = (2*n + 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-73c89ff38d5bbc45
• 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-73c89ff38d5bbc45
• 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-73c89ff38d5bbc45

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-73c89ff38d5bbc45

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

  • premise (sequence): Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-61ffea65f7eec286 root: theorem check

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

    • basis_map A007318 e-83332bc01455e815

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers root_domain e-8128a83cc2062067 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-7d7235cf57d80f69

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers root_domain e-8128a83cc2062067 (expanded above)
    • parent_provers root_domain e-bf1e7898c429e3eb root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image affine, shift 0 e-ab4aac9d5a3bd564

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers root_domain e-8128a83cc2062067 (expanded above)
        • ↵ preserver_image mobius, shift 0 e-7d7235cf57d80f69 (expanded above)

Related sequences ?

Same fingerprint (19)

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

• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A178822 → basis A008277
• A028326 → basis A008277
• A178821 → basis A008277
• A085881 → basis A008277
• A135838 → basis A008277
• A178820 → basis A008277
• A038325 → basis A008277
• A085880 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• basis-map coefficients of A113214, shifted → basis A008277
• A156992 → basis A008277
• A143494 → reverse
• A126351 → reverse
• A038338 → basis A008277
• A356546 → basis A008277
• A344565 → basis A008277
• A038208 → basis A008277
• A038221 → basis A008277

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