Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.

OEIS matches ?

• A003581: 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:
   1: 1,          11,            1;
   2: 1,         111,           30,             1;
   3: 1,        1111,          681,            58,            1;
   4: 1,       11111,        14050,          2305,           95,            1;
   5: 1,      111111,       278061,         78590,         5820,          141,           1;
   6: 1,     1111111,      5394270,       2478581,       293930,        12306,         196,         1;
   7: 1,    11111111,    103602241,      74794538,     13353991,       860006,       23086,       260,       1;
   8: 1,   111111111,   1979553690,    2197849305,    568892205,     52914267,     2129736,     39726,     333,     1;
   9: 1,  1111111111,  37722631221,   63519334230,  23246860890,   3002948487,   170049747,   4672200,   64035,   415,   1;
  10: 1, 11111111111, 717841104310, 1816263989661, 923653187160, 161382491292, 12355684572, 469070547, 9346755, 98065, 506, 1;

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 11*x + x^2.

P[1, x_] := 1 + 11*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 11*x + x^2; for(n = 2, N, my(i = n); v[i] = (1 + 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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-b6e553c117a424c0
• Coefficients of one sign. Every nonzero row, n >= 1: 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-b6e553c117a424c0
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-b6e553c117a424c0
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-b6e553c117a424c0
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 1] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 1] 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-6b24216ed39c680d

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-6b24216ed39c680d root: theorem check

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

• theorem_check liu_wang e-b6e553c117a424c0 root: theorem check

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

Used in 2 proof steps

• e-a20ccc97cc2b4079 obreschkoff_combination → obreschkoff_combination of Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - …
• e-a20ccc97cc2b4079 obreschkoff_combination → obreschkoff_combination of Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - …

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