A282629: Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].

OEIS matches ?

• A282629: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A284859: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1;
  1: 1,      3;
  2: 1,     15,        9;
  3: 1,     63,      108,        27;
  4: 1,    255,      945,       594,        81;
  5: 1,   1023,     7380,      8775,      2835,       243;
  6: 1,   4095,    54729,    109890,     63180,     12393,      729;
  7: 1,  16383,   395388,   1263087,   1151010,    387828,    51030,     2187;
  8: 1,  65535,  2816865,  13817034,  18752391,   9658278,  2133054,   201204,   6561;
  9: 1, 262143, 19914660, 146620935, 285232185, 210789621, 69502860, 10825650, 767637, 19683;

The row polynomials satisfy P(n,x) = (1+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 1, x] + 3*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 3*x)*v[i - 1] + 3*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)

Triangle begins:
  0: 1;
  1: 1,      3;
  2: 1,     15,        9;
  3: 1,     63,      108,        27;
  4: 1,    255,      945,       594,        81;
  5: 1,   1023,     7380,      8775,      2835,       243;
  6: 1,   4095,    54729,    109890,     63180,     12393,      729;
  7: 1,  16383,   395388,   1263087,   1151010,    387828,    51030,     2187;
  8: 1,  65535,  2816865,  13817034,  18752391,   9658278,  2133054,   201204,   6561;
  9: 1, 262143, 19914660, 146620935, 285232185, 210789621, 69502860, 10825650, 767637, 19683;

The row polynomials satisfy P(n,x) = (1+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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-949f12f88dca835e
• 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-949f12f88dca835e
• 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-949f12f88dca835e
• 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-949f12f88dca835e
• 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-bb1d1ed479feaa5d

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-bb1d1ed479feaa5d root: theorem check

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

• theorem_check liu_wang e-949f12f88dca835e root: theorem check

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

Used in 2 proof steps

• e-b6db328b817fb97a obreschkoff_combination → obreschkoff_combination of Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].
• e-b6db328b817fb97a obreschkoff_combination → obreschkoff_combination of Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].

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