A128813 → basis A013611

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,            4;
  2:           8,           40,            48;
  3:          56,          472,          1296,           1152;
  4:         616,         7432,         33136,          64512,           46080;
  5:        9856,       155872,        976096,        3020352,         4608000,         2764800;
  6:      216832,      4257088,      34567360,      148439808,       355085568,       447897600,        232243200;
  7:     6288128,    147740736,    1479247296,     8176298752,     26922739968,     52758614016,      56899584000,      26011238400;
  8:   232660736,   6371897664,   76006815936,   515534664448,   2173528399104,   5828943273984,    9702525026304,    9155955916800,    3745618329600;
  9: 10702393856, 334986225024, 4643255112576, 37395821433088, 192778545959424, 659366502441984, 1495525940527104, 2167628476907520, 1820370508185600, 674211299328000;

The row polynomials satisfy P(n,x) = ((1/2)*n^2+(1/2)*n+1+(2*n^2+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 + (2*n^2 + 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 + (2*n^2 + 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-7d5e908b514e0d51
• 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-7d5e908b514e0d51
• 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-7d5e908b514e0d51
• 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-7d5e908b514e0d51

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 A013611 e-7d5e908b514e0d51

  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