Stirling2 triangle with scaled diagonals (powers of 6).

OEIS matches ?

• A005012: 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:        36,           18,            1;
  1:       216,          252,           36,             1;
  2:      1296,         3240,          900,            60,            1;
  3:      7776,        40176,        19440,          2340,           90,           1;
  4:     46656,       489888,       390096,         75600,         5040,         126,          1;
  5:    279936,      5925312,      7511616,       2204496,       226800,        9576,        168,         1;
  6:   1679616,     71383680,    141134400,      60419520,      9008496,      571536,      16632,       216,       1;
  7:  10077696,    858283776,   2611802880,    1591202880,    330674400,    29583792,    1270080,     27000,     270,     1;
  8:  60466176,  10309483008,  47870735616,   40800672000,  11511434880,  1395690912,   82927152,   2566080,   41580,   330,   1;
  9: 362797056, 123774262272, 871982724096, 1027086863616, 386143718400, 61756307712, 4878631296, 206098992, 4811400, 61380, 396, 1;

The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 36 + 18*x + x^2.

P[0, x_] := 36 + 18*x + x^2;
P[n_, x_] := P[n, x] = Expand[(6 + x)*P[n - 1, x] + 6*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] = 36 + 18*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + x)*v[i - 1] + 6*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-87ce0e9883ea7bf4
• 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-87ce0e9883ea7bf4
• 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-87ce0e9883ea7bf4
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-87ce0e9883ea7bf4
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 6] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 6] with m = 0, target (-inf, 0]
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-72753d30bc49949c

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_theorem e-72753d30bc49949c root: theorem check

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

• theorem_check liu_wang e-87ce0e9883ea7bf4 root: theorem check

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

Used in 2 proof steps

• e-288a11b1cf9253fe obreschkoff_combination → obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 6).
• e-288a11b1cf9253fe obreschkoff_combination → obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 6).

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