A051339: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 6.

OEIS matches ?

• A051339: triangle, 45 terms agree, 9 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:          -7,          1;
  2:          56,        -15,          1;
  3:        -504,        191,        -24,         1;
  4:        5040,      -2414,        431,       -34,         1;
  5:      -55440,      31594,      -7155,       805,       -45,       1;
  6:      665280,    -434568,     117454,    -16815,      1345,     -57,       1;
  7:    -8648640,    6314664,   -1961470,    336049,    -34300,    2086,     -70,    1;
  8:   121080960,  -97053936,   33775244,  -6666156,    816249,  -63504,    3066,  -84,   1;
  9: -1816214400, 1576890000, -603682596, 133767584, -18909891, 1768809, -109494, 4326, -99, 1;

The row polynomials satisfy P(n,x) = -(n+6-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 + 6 - 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 + 6 - 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:          -7,          1;
  2:          56,        -15,          1;
  3:        -504,        191,        -24,         1;
  4:        5040,      -2414,        431,       -34,         1;
  5:      -55440,      31594,      -7155,       805,       -45,       1;
  6:      665280,    -434568,     117454,    -16815,      1345,     -57,       1;
  7:    -8648640,    6314664,   -1961470,    336049,    -34300,    2086,     -70,    1;
  8:   121080960,  -97053936,   33775244,  -6666156,    816249,  -63504,    3066,  -84,   1;
  9: -1816214400, 1576890000, -603682596, 133767584, -18909891, 1768809, -109494, 4326, -99, 1;

The row polynomials satisfy P(n,x) = -(n+6-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 + 6 - 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 + 6 - 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 [13/2, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-297f51c747adaedd
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) 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-297f51c747adaedd
• 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-297f51c747adaedd

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-297f51c747adaedd

  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-e24e4dcabf4aa882 obreschkoff_combination → obreschkoff_combination of Generalized Stirling number triangle of first kind read by row…
• e-e24e4dcabf4aa882 obreschkoff_combination → obreschkoff_combination of Generalized Stirling number triangle of first kind read by row…

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