A225469: Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

OEIS matches ?

• A355167: 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 ?
• A225469: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 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:
   1:     1;
   2:     3,        1;
   3:     9,       10,        1;
   4:    27,       79,       21,        1;
   5:    81,      580,      310,       36,        1;
   6:   243,     4141,     3990,      850,       55,       1;
   7:   729,    29230,    48031,    16740,     1895,      78,      1;
   8:  2187,   205339,   557571,   299131,    52745,    3689,    105,     1;
   9:  6561,  1439560,  6338620,  5044536,  1301286,  137592,   6524,   136,   1;
  10: 19683, 10083481, 71164380, 82006660, 29768970, 4465902, 313740, 10740, 171, 1;

The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + 4*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; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + 4*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)

Triangle begins:
  0:     1;
  1:     3,        1;
  2:     9,       10,        1;
  3:    27,       79,       21,        1;
  4:    81,      580,      310,       36,        1;
  5:   243,     4141,     3990,      850,       55,       1;
  6:   729,    29230,    48031,    16740,     1895,      78,      1;
  7:  2187,   205339,   557571,   299131,    52745,    3689,    105,     1;
  8:  6561,  1439560,  6338620,  5044536,  1301286,  137592,   6524,   136,   1;
  9: 19683, 10083481, 71164380, 82006660, 29768970, 4465902, 313740, 10740, 171, 1;

The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + 4*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[(3 + x)*P[n - 1, x] + 4*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] = (3 + x)*v[i - 1] + 4*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 >= 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-8ea25153a52b5bd5
• 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-8ea25153a52b5bd5
• 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-8ea25153a52b5bd5
• 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-8ea25153a52b5bd5
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 3] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 3] 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-64501166ff7efe7d

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-64501166ff7efe7d root: theorem check

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

• theorem_check liu_wang e-8ea25153a52b5bd5 root: theorem check

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

Used in 2 proof steps

• e-1f2ac171cb8f6597 obreschkoff_combination → obreschkoff_combination of Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirl…
• e-1f2ac171cb8f6597 obreschkoff_combination → obreschkoff_combination of Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirl…

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