A094645: 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), where Stirling1 is A048994.

OEIS matches ?

• A094645: 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 ?

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,     1;
  2:  0,    -1,     1;
  3:  0,    -1,     0,   1;
  4:  0,    -2,    -1,   2,    1;
  5:  0,    -6,    -5,   5,    5,    1;
  6:  0,   -24,   -26,  15,   25,    9,    1;
  7:  0,  -120,  -154,  49,  140,   70,   14,   1;
  8:  0,  -720, -1044, 140,  889,  560,  154,  20,  1;
  9:  0, -5040, -8028, -64, 6363, 4809, 1638, 294, 27, 1;

The row polynomials satisfy P(n,x) = (n-2+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 - 2 + 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 - 2 + 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: -1,     1;
  2:  0,    -1,     1;
  3:  0,    -1,     0,   1;
  4:  0,    -2,    -1,   2,    1;
  5:  0,    -6,    -5,   5,    5,    1;
  6:  0,   -24,   -26,  15,   25,    9,    1;
  7:  0,  -120,  -154,  49,  140,   70,   14,   1;
  8:  0,  -720, -1044, 140,  889,  560,  154,  20,  1;
  9:  0, -5040, -8028, -64, 6363, 4809, 1638, 294, 27, 1;

The row polynomials satisfy P(n,x) = (n-2+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 - 2 + 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 - 2 + 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, 3/2]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-0658035b499d17ce
• 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-0658035b499d17ce
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-2866478407d5f715

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 A130595 e-2866478407d5f715

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

      Parent-row provers of polytool polytool ?

• preserver_image affine, shift 0 e-0658035b499d17ce

  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-c8fbb9d7e751998c obreschkoff_combination → obreschkoff_combination of Triangle of generalized Stirling numbers of the first kind rea…
• e-c8fbb9d7e751998c 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