A128813 → basis A013610

OEIS matches ?

• A277347: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?

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:           2,            3;
  2:           8,           30,            27;
  3:          56,          354,           729,            486;
  4:         616,         5574,         18639,          27216,          14580;
  5:        9856,       116904,        549054,        1274211,        1458000,          656100;
  6:      216832,      3192816,      19444140,       62623044,      112351293,       106288200,        41334300;
  7:     6288128,    110805552,     832076604,     3449376036,     8518523193,     12519866412,     10126903500,      3472081200;
  8:   232660736,   4778923248,   42753833964,   217491186564,   687717970029,   1383235562088,   1726841001996,   1222172582400,    374984769600;
  9: 10702393856, 251239668768, 2611831000824, 15776362167084, 60996336807474, 156470761809963, 266171486973696, 289343474059860, 182242598025600, 50622943896000;

The row polynomials satisfy P(n,x) = ((1/2)*n^2+(1/2)*n+1+((3/2)*n^2+(3/2)*n)*x)*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 + ((3/2)*n^2 + (3/2)*n)*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; for(n = 1, N, my(i = n + 1); v[i] = ((1/2)*n^2 + (1/2)*n + 1 + ((3/2)*n^2 + (3/2)*n)*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-201bedfe8f6594c1
• 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-201bedfe8f6594c1
• 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-201bedfe8f6594c1
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-201bedfe8f6594c1

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-201bedfe8f6594c1

  Geometric basis map (elementary) polytool ?

  • 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