A154986 → basis A007318

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,        6,        1;
  3:      24,       36,       14,        1;
  4:     120,      240,      148,       28,        1;
  5:     720,     1800,     1536,      504,       50,       1;
  6:    5040,    15120,    16512,     7824,     1474,      82,      1;
  7:   40320,   141120,   188256,   117840,    34040,    3780,    126,     1;
  8:  362880,  1451520,  2289024,  1786752,   706608,  128736,   8680,   184,   1;
  9: 3628800, 16329600, 29744640, 27901440, 14135328, 3687120, 427328, 18192, 258, 1;

The row polynomials satisfy P(n,x) = (n^2-n+(n^2-n)*x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(n^2 - n + (n^2 - n)*x)*P[n - 2, x] + (2 + 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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (n^2 - n + (n^2 - n)*x)*v[i - 2] + (2 + 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-d0341b4ac1663435
• 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-d0341b4ac1663435, e-fea99146cab206a8
• 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-d0341b4ac1663435, e-fea99146cab206a8
• 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-d0341b4ac1663435, e-fea99146cab206a8
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-fea99146cab206a8

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-d0341b4ac1663435

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-bad4de7d401fa939 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

    • preserver_image reversal, shift 0 e-3d59ef840ca39edf

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang_derivative_lag e-bad4de7d401fa939 (expanded above)
• preserver_image affine, shift 0 e-fea99146cab206a8

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check liu_wang_derivative_lag e-bad4de7d401fa939 (expanded above)
    • ↵ preserver_image reversal, shift 0 e-3d59ef840ca39edf (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A154986 → x ↦ x/(x+1)

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