A290595: Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stirling 1), for n >= 0.

OEIS matches ?

• A290595: 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 ?
• A011781: 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:
   1:         1;
   2:         1,           2;
   3:         4,          19,            4;
   4:        28,         222,          147,            8;
   5:       280,        3194,         4128,          887,           16;
   6:      3640,       55024,       113566,        52538,         4835,          32;
   7:     58240,     1107336,      3268788,      2562676,       555684,       25167,         64;
   8:   1106560,    25526192,    100544412,    117517960,     45415640,     5301150,     128203,       128;
   9:  24344320,   663605680,   3325767376,   5352311764,   3189383200,   695714590,   47537320,    646519,     256;
  10: 608608000, 19213911360, 118361719296, 248493947496, 208996478388, 72479948400, 9696965250, 410038434, 3245139, 512;

The row polynomials satisfy P(n,x) = (3*n-5+(3*n-4)*x)*P(n-1,x) + (3*x-3*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*n - 5 + (3*n - 4)*x)*P[n - 1, x] + (3*x - 3*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*n - 5 + (3*n - 4)*x)*v[i - 1] + (3*x - 3*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:         1,           2;
  2:         4,          19,            4;
  3:        28,         222,          147,            8;
  4:       280,        3194,         4128,          887,           16;
  5:      3640,       55024,       113566,        52538,         4835,          32;
  6:     58240,     1107336,      3268788,      2562676,       555684,       25167,         64;
  7:   1106560,    25526192,    100544412,    117517960,     45415640,     5301150,     128203,       128;
  8:  24344320,   663605680,   3325767376,   5352311764,   3189383200,   695714590,   47537320,    646519,     256;
  9: 608608000, 19213911360, 118361719296, 248493947496, 208996478388, 72479948400, 9696965250, 410038434, 3245139, 512;

The row polynomials satisfy P(n,x) = (3*n-2+(3*n-1)*x)*P(n-1,x) + (3*x-3*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*n - 2 + (3*n - 1)*x)*P[n - 1, x] + (3*x - 3*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*n - 2 + (3*n - 1)*x)*v[i - 1] + (3*x - 3*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 (-inf, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-8066ebfb4deadcc4
• 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-8066ebfb4deadcc4
• 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-8066ebfb4deadcc4
• 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-8066ebfb4deadcc4

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• theorem_check liu_wang e-8066ebfb4deadcc4 root: theorem check

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

Used in 2 proof steps

• e-35685a58b6434e5d obreschkoff_combination → obreschkoff_combination of Triangle T(n, k) read by rows: row n gives the coefficients of…
• e-35685a58b6434e5d obreschkoff_combination → obreschkoff_combination of Triangle T(n, k) read by rows: row n gives the coefficients of…

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