A049459: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 3.

OEIS matches ?

• A049459: triangle, 41 terms agree, 8 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A105752: 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:        -4,         1;
  2:        20,        -9,         1;
  3:      -120,        74,       -15,        1;
  4:       840,      -638,       179,      -22,        1;
  5:     -6720,      5944,     -2070,      355,      -30,      1;
  6:     60480,    -60216,     24574,    -5265,      625,    -39,      1;
  7:   -604800,    662640,   -305956,    77224,   -11515,   1015,    -49,    1;
  8:   6652800,  -7893840,   4028156, -1155420,   203889, -22680,   1554,  -60,   1;
  9: -79833600, 101378880, -56231712, 17893196, -3602088, 476049, -41328, 2274, -72, 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:        -4,         1;
  2:        20,        -9,         1;
  3:      -120,        74,       -15,        1;
  4:       840,      -638,       179,      -22,        1;
  5:     -6720,      5944,     -2070,      355,      -30,      1;
  6:     60480,    -60216,     24574,    -5265,      625,    -39,      1;
  7:   -604800,    662640,   -305956,    77224,   -11515,   1015,    -49,    1;
  8:   6652800,  -7893840,   4028156, -1155420,   203889, -22680,   1554,  -60,   1;
  9: -79833600, 101378880, -56231712, 17893196, -3602088, 476049, -41328, 2274, -72, 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 [7/2, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-c581ca12cb2a386a
• 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-c581ca12cb2a386a
• 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-c581ca12cb2a386a

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-c581ca12cb2a386a

  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-d08409290ded4df6 obreschkoff_combination → obreschkoff_combination of Generalized Stirling number triangle of first kind read by row…
• e-d08409290ded4df6 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