A128813 → 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:           8,            13,              3;
  3:          56,           217,            135,              18;
  4:         616,          5117,           6355,            2088,             180;
  5:        9856,        167867,         369085,          222693,           45000,           2700;
  6:      216832,       7425257,       27146647,        26679690,         9446553,        1287900,          56700;
  7:     6288128,     429310945,     2515372191,      3774911086,      2078995293,      482158584,       47231100,       1587600;
  8:   232660736,   31566071593,   289630762839,    637915506346,    512194947129,   179472243276,    29307177324,    2159136000,     57153600;
  9: 10702393856, 2882982248083, 40810256967789, 128496090976381, 144462255836724, 71685750548601, 17337318981804, 2098271075580, 120365481600, 2571912000;

The row polynomials satisfy P(n,x) = ((1/2)*n^2+(1/2)*n+1+((1/2)*n^2+(1/2)*n)*x)*P(n-1,x) + ((1/2)*n^2+(1/2)*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[((1/2)*n^2 + (1/2)*n + 1 + ((1/2)*n^2 + (1/2)*n)*x)*P[n - 1, x] + ((1/2)*n^2 + (1/2)*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] = ((1/2)*n^2 + (1/2)*n + 1 + ((1/2)*n^2 + (1/2)*n)*x)*v[i - 1] + ((1/2)*n^2 + (1/2)*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-cebfd0c72bfddbbf
• 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-cebfd0c72bfddbbf
• 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-cebfd0c72bfddbbf

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-cebfd0c72bfddbbf

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

  • premise (sequence): Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-fd268ef2c64764c5 root: theorem check

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

    • parent_provers paper_theorem e-a464fe87e26690eb root: theorem check

      Parent-row provers of polytool polytool ?

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