A391533 → x ↦ x+1

OEIS matches ?

• A183355: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A188825: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A257970: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:    9,     6,     1;
  1:   18,    21,     8,     1;
  2:   36,    60,    37,    10,     1;
  3:   72,   156,   134,    57,    12,     1;
  4:  144,   384,   424,   248,    81,    14,     1;
  5:  288,   912,  1232,   920,   410,   109,    16,    1;
  6:  576,  2112,  3376,  3072,  1740,   628,   141,   18,    1;
  7: 1152,  4800,  8864,  9520,  6552,  2996,   910,  177,   20,   1;
  8: 2304, 10752, 22528, 27904, 22624, 12544,  4816, 1264,  217,  22,  1;
  9: 4608, 23808, 55808, 78336, 73152, 47712, 22176, 7344, 1698, 261, 24, 1;

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 6*x + x^2.

P[0, x_] := 9 + 6*x + x^2;
P[n_, x_] := P[n, x] = Expand[(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] = 9 + 6*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (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-98125ea59568ff18
• 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-3fc4d53dea6f488e, e-98125ea59568ff18
• 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-3fc4d53dea6f488e, e-98125ea59568ff18
• 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-3fc4d53dea6f488e
• 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-3fc4d53dea6f488e

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-98125ea59568ff18

  Geometric basis map (elementary) polytool ?

  • premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check

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

    • parent_provers periodic_product e-5e5f630c1e2baa8a root: theorem check

      Parent-row provers of polytool polytool ?

• preserver_image affine, shift 0 e-3fc4d53dea6f488e

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check liu_wang e-3cff49c1888b5e80 (expanded above)
    • ↵ parent_provers periodic_product e-5e5f630c1e2baa8a (expanded above)

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