A094646: Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n, m) * binomial(m, k) * 2^(m-k), where Stirling1 is A048994.

OEIS matches ?

• A014040: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
• A014032: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
• A016399: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
• A014038: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
• A016383: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
• A014026: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
• A016411: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS
• A094646: triangle, 66 terms agree, 11 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A014028: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS

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: -2,    1;
  2:  2,   -3,    1;
  3:  0,    2,   -3,     1;
  4:  0,    2,   -1,    -2,     1;
  5:  0,    4,    0,    -5,     0,    1;
  6:  0,   12,    4,   -15,    -5,    3,   1;
  7:  0,   48,   28,   -56,   -35,    7,   7,   1;
  8:  0,  240,  188,  -252,  -231,    0,  42,  12,  1;
  9:  0, 1440, 1368, -1324, -1638, -231, 252, 114, 18, 1;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n - 3 + x)*P[n - 1, 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] = (n - 3 + x)*v[i - 1]); 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: -2,    1;
  2:  2,   -3,    1;
  3:  0,    2,   -3,     1;
  4:  0,    2,   -1,    -2,     1;
  5:  0,    4,    0,    -5,     0,    1;
  6:  0,   12,    4,   -15,    -5,    3,   1;
  7:  0,   48,   28,   -56,   -35,    7,   7,   1;
  8:  0,  240,  188,  -252,  -231,    0,  42,  12,  1;
  9:  0, 1440, 1368, -1324, -1638, -231, 252, 114, 18, 1;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n - 3 + x)*P[n - 1, 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] = (n - 3 + x)*v[i - 1]); 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, 5/2]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-2198daa4fdd0bfd0
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-2198daa4fdd0bfd0

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

• preserver_image affine, shift 0 e-2198daa4fdd0bfd0

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-59b849691f316c6f root: theorem check

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

    • parent_provers root_domain e-ef36674c01230a86 root: theorem check

      Parent-row provers of polytool polytool ?

Used in 2 proof steps

• e-1e544bff83909e45 obreschkoff_combination → obreschkoff_combination of Triangle of generalized Stirling numbers of the first kind rea…
• e-1e544bff83909e45 obreschkoff_combination → obreschkoff_combination of Triangle of generalized Stirling numbers of the first kind rea…

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