Stirling2 triangle with scaled diagonals (powers of 9).

OEIS matches ?

• A075508: 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:          81,            27,              1;
  1:         729,           567,             54,              1;
  2:        6561,         10935,           2025,             90,             1;
  3:       59049,        203391,          65610,           5265,           135,            1;
  4:      531441,       3720087,        1974861,         255150,         11340,          189,           1;
  5:     4782969,      67493007,       57041334,       11160261,        765450,        21546,         252,          1;
  6:    43046721,    1219657095,     1607609025,      458810730,      45605511,      1928934,       37422,        324,        1;
  7:   387420489,   21996874431,    44625100770,    18124795305,    2511058725,    149767947,     4286520,      60750,      405,      1;
  8:  3486784401,  396331160247,  1226874595221,   697117731750,  131122437930,  10598527863,   419818707,    8660520,    93555,    495,   1;
  9: 31381059609, 7137447668847, 33521945231214, 26323112938221, 6597627438600, 703442942532, 37047106404, 1043376147, 16238475, 138105, 594, 1;

The row polynomials satisfy P(n,x) = (9+x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 81 + 27*x + x^2.

P[0, x_] := 81 + 27*x + x^2;
P[n_, x_] := P[n, x] = Expand[(9 + x)*P[n - 1, x] + 9*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] = 81 + 27*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (9 + x)*v[i - 1] + 9*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-b7566ed62ccee4c1
• 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-b7566ed62ccee4c1
• 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-b7566ed62ccee4c1
• 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-b7566ed62ccee4c1
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 9] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 9] 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-25f661182ec3ab36

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-25f661182ec3ab36 root: theorem check

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

• theorem_check liu_wang e-b7566ed62ccee4c1 root: theorem check

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

Used in 2 proof steps

• e-cdccc64d5381f9ff obreschkoff_combination → obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 9).
• e-cdccc64d5381f9ff obreschkoff_combination → obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 9).

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