Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

OEIS matches ?

• A075506: 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:         49,           21,             1;
  1:        343,          343,            42,             1;
  2:       2401,         5145,          1225,            70,             1;
  3:      16807,        74431,         30870,          3185,           105,            1;
  4:     117649,      1058841,        722701,        120050,          6860,          147,           1;
  5:     823543,     14941423,      16235562,       4084101,        360150,        13034,         196,         1;
  6:    5764801,    210003465,     355888225,     130590390,      16689351,       907578,       22638,       252,       1;
  7:   40353607,   2945813311,    7683656190,    4012419145,     714717675,     54807627,     2016840,     36750,     315,     1;
  8:  282475249,  41281739961,  164302593301,  120031392250,   29027537770,   3016638009,   153632787,   4074840,   56595,   385,   1;
  9: 1977326743, 578226834703, 3491636199282, 3525181576301, 1135995214200, 155726334148, 10544644572, 381823827, 7640325, 83545, 462, 1;

The row polynomials satisfy P(n,x) = (7+x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 49 + 21*x + x^2.

P[0, x_] := 49 + 21*x + x^2;
P[n_, x_] := P[n, x] = Expand[(7 + x)*P[n - 1, x] + 7*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] = 49 + 21*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + x)*v[i - 1] + 7*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-4e6146a9e968e4df
• 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-4e6146a9e968e4df
• 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-4e6146a9e968e4df
• 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-4e6146a9e968e4df
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 7] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 7] 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-d44cd479ca8327b0

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-d44cd479ca8327b0 root: theorem check

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

• theorem_check liu_wang e-4e6146a9e968e4df root: theorem check

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

Used in 2 proof steps

• e-a22896546ec845f5 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: Stirling2 triangle with scaled diagonal…
• e-a22896546ec845f5 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: Stirling2 triangle with scaled diagonal…

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