Stirling2 triangle with scaled diagonals (powers of 5).

OEIS matches ?

• A005011: 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:       25,          15,            1;
  1:      125,         175,           30,            1;
  2:      625,        1875,          625,           50,            1;
  3:     3125,       19375,        11250,         1625,           75,           1;
  4:    15625,      196875,       188125,        43750,         3500,         105,          1;
  5:    78125,     1984375,      3018750,      1063125,       131250,        6650,        140,        1;
  6:   390625,    19921875,     47265625,     24281250,      4344375,      330750,      11550,      180,       1;
  7:  1953125,   199609375,    728906250,    532890625,    132890625,    14266875,     735000,    18750,     225,     1;
  8:  9765625,  1998046875,  11133203125,  11386718750,   3855156250,   560896875,   39991875,  1485000,   28875,   275,   1;
  9: 48828125, 19990234375, 168996093750, 238867578125, 107765625000, 20682062500, 1960612500, 99391875, 2784375, 42625, 330, 1;

The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 25 + 15*x + x^2.

P[0, x_] := 25 + 15*x + x^2;
P[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + 5*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] = 25 + 15*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1] + 5*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-42338bfb8394e6f8
• 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-42338bfb8394e6f8
• 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-42338bfb8394e6f8
• 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-42338bfb8394e6f8
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 5] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 5] 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-56cf6b2b70702390

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-56cf6b2b70702390 root: theorem check

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

• theorem_check liu_wang e-42338bfb8394e6f8 root: theorem check

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

Used in 2 proof steps

• e-206745053438cac1 obreschkoff_combination → obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 5).
• e-206745053438cac1 obreschkoff_combination → obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 5).

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