Stirling2 triangle with scaled diagonals (powers of 10).

OEIS matches ?

• A075509: 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:          100,             30,              1;
  1:         1000,            700,             60,              1;
  2:        10000,          15000,           2500,            100,              1;
  3:       100000,         310000,          90000,           6500,            150,             1;
  4:      1000000,        6300000,        3010000,         350000,          14000,           210,           1;
  5:     10000000,      127000000,       96600000,       17010000,        1050000,         26600,         280,          1;
  6:    100000000,     2550000000,     3025000000,      777000000,       69510000,       2646000,       46200,        360,        1;
  7:   1000000000,    51100000000,    93300000000,    34105000000,     4252500000,     228270000,     5880000,      75000,      450,      1;
  8:  10000000000,  1023000000000,  2850100000000,  1457500000000,   246730000000,   17948700000,   639870000,   11880000,   115500,    550,   1;
  9: 100000000000, 20470000000000, 86526000000000, 61150100000000, 13794000000000, 1323652000000, 62739600000, 1590270000, 22275000, 170500, 660, 1;

The row polynomials satisfy P(n,x) = (10+x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 100 + 30*x + x^2.

P[0, x_] := 100 + 30*x + x^2;
P[n_, x_] := P[n, x] = Expand[(10 + x)*P[n - 1, x] + 10*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] = 100 + 30*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + x)*v[i - 1] + 10*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-b1938fcbfe117d00
• 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-b1938fcbfe117d00
• 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-b1938fcbfe117d00
• 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-b1938fcbfe117d00
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 10] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 10] 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-506dd990cfc1b4f3

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-506dd990cfc1b4f3 root: theorem check

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

• theorem_check liu_wang e-b1938fcbfe117d00 root: theorem check

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

Used in 2 proof steps

• e-f660dc27e1b4f4f9 obreschkoff_combination → obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 10).
• e-f660dc27e1b4f4f9 obreschkoff_combination → obreschkoff_combination of Stirling2 triangle with scaled diagonals (powers of 10).

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