A048176: Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).

OEIS matches ?

• A048176: triangle, 38 terms agree, 8 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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:              -10,               1;
  2:              200,             -30,               1;
  3:            -6000,            1100,             -60,            1;
  4:           240000,          -50000,            3500,         -100,            1;
  5:        -12000000,         2740000,         -225000,         8500,         -150,         1;
  6:        720000000,      -176400000,        16240000,      -735000,        17500,      -210,        1;
  7:     -50400000000,     13068000000,     -1313200000,     67690000,     -1960000,     32200,     -280,     1;
  8:    4032000000000,  -1095840000000,    118124000000,  -6728400000,    224490000,  -4536000,    54600,  -360,    1;
  9: -362880000000000, 102657600000000, -11727000000000, 723680000000, -26932500000, 632730000, -9450000, 87000, -450, 1;

The row polynomials satisfy P(n,x) = -(10*n-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(10*n - 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] = -(10*n - 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:
   1:                1;
   2:              -10,               1;
   3:              200,             -30,               1;
   4:            -6000,            1100,             -60,            1;
   5:           240000,          -50000,            3500,         -100,            1;
   6:        -12000000,         2740000,         -225000,         8500,         -150,         1;
   7:        720000000,      -176400000,        16240000,      -735000,        17500,      -210,        1;
   8:     -50400000000,     13068000000,     -1313200000,     67690000,     -1960000,     32200,     -280,     1;
   9:    4032000000000,  -1095840000000,    118124000000,  -6728400000,    224490000,  -4536000,    54600,  -360,    1;
  10: -362880000000000, 102657600000000, -11727000000000, 723680000000, -26932500000, 632730000, -9450000, 87000, -450, 1;

The row polynomials satisfy P(n,x) = -(10*n-10-x)*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(10*n - 10 - x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(10*n - 10 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [5, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-040c826eba4e7d74
• 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-040c826eba4e7d74
• 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-040c826eba4e7d74

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-040c826eba4e7d74

  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-cdcdb3aca64a476d obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 a…
• e-cdcdb3aca64a476d obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 a…

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