A225473: Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

OEIS matches ?

• A355220: 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 ?
• A225473: triangle, 37 terms agree, 8 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,        4;
   3:     9,       40,         32;
   4:    27,      316,        672,         384;
   5:    81,     2320,       9920,       13824,         6144;
   6:   243,    16564,     127680,      326400,       337920,       122880;
   7:   729,   116920,    1536992,     6428160,     11642880,      9584640,      2949120;
   8:  2187,   821356,   17842272,   114866304,    324065280,    453304320,    309657600,     82575360;
   9:  6561,  5758240,  202835840,  1937101824,   7995101184,  16907304960,  19240058880,  11230248960,   2642411520;
  10: 19683, 40333924, 2277260160, 31490557440, 182900551680, 548770037760, 925256908800, 886859366400, 451852369920, 95126814720;

The row polynomials satisfy P(n,x) = (3+4*x)*P(n-1,x) + (4*x+4*x^2)*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 + 4*x)*P[n - 1, x] + (4*x + 4*x^2)*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 + 4*x)*v[i - 1] + (4*x + 4*x^2)*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,        4;
  2:     9,       40,         32;
  3:    27,      316,        672,         384;
  4:    81,     2320,       9920,       13824,         6144;
  5:   243,    16564,     127680,      326400,       337920,       122880;
  6:   729,   116920,    1536992,     6428160,     11642880,      9584640,      2949120;
  7:  2187,   821356,   17842272,   114866304,    324065280,    453304320,    309657600,     82575360;
  8:  6561,  5758240,  202835840,  1937101824,   7995101184,  16907304960,  19240058880,  11230248960,   2642411520;
  9: 19683, 40333924, 2277260160, 31490557440, 182900551680, 548770037760, 925256908800, 886859366400, 451852369920, 95126814720;

The row polynomials satisfy P(n,x) = (3+4*x)*P(n-1,x) + (4*x+4*x^2)*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 + 4*x)*P[n - 1, x] + (4*x + 4*x^2)*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 + 4*x)*v[i - 1] + (4*x + 4*x^2)*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 (-1, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-a7b6cc9ddce470c2
• 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-a7b6cc9ddce470c2
• 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-a7b6cc9ddce470c2
• 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-a7b6cc9ddce470c2
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [-1, 3] to roots in [-1, 0]: source [-1, 3], target [-1, 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-9b2a752e203a2a8f

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-9b2a752e203a2a8f root: theorem check

  Interval basis map theorem [MW08, Thm 2, Cor 3] polytool RealRooted intervalBasisMap_maps_realRootedIn_iff ?

• theorem_check ma_wang_interval e-a7b6cc9ddce470c2 root: theorem check

  Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 2 proof steps

• e-d4e2c570702f66e5 obreschkoff_combination → obreschkoff_combination of Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the St…
• e-d4e2c570702f66e5 obreschkoff_combination → obreschkoff_combination of Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the St…

Related sequences ?

No candidate equalities or related sequences.

References

1. [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