Real-rooted atlas

A008276: Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n)/x in powers of x.

A059364 → x ↦ −x−1 (shift -1)

A008276 only rational roots

Proved for all n: all roots real, in [−1, ∞) · interlacing (weak) what this means

Recurrence from OEIS A008276
\(P_{n}(x) = \left(1 + x - n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
21-10
31-320
41-611-60
51-1035-50240
61-1585-225274-1200
71-21175-7351624-17647200
81-28322-19606769-1313213068-50400

Show all 15 stored rows

Generated b-files: A008275 (reversed triangle), A008276 ?

Computed in your browser ?

Technical details
Node id
n-a6c3537128a476c6 · revision 1 · JSON
Definition source
oeis_catalog:A008276
Fingerprint
f6-ff058243d1bd7929

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

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.

The atlas's own rows: n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  -1;
   3: 1,  -3,   2;
   4: 1,  -6,  11,    -6;
   5: 1, -10,  35,   -50,    24;
   6: 1, -15,  85,  -225,   274,    -120;
   7: 1, -21, 175,  -735,  1624,   -1764,    720;
   8: 1, -28, 322, -1960,  6769,  -13132,  13068,    -5040;
   9: 1, -36, 546, -4536, 22449,  -67284, 118124,  -109584,   40320;
  10: 1, -45, 870, -9450, 63273, -269325, 723680, -1172700, 1026576, -362880;
%F Formula
The row polynomials satisfy P(n,x) = (1-(n-1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 - (n - 1)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 - (n - 1)*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, v[i])))};
T(10)

Generated with polytool c48629374597.

A008275: reversed triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1:       1;
   2:      -1,       1;
   3:       2,      -3,        1;
   4:      -6,      11,       -6,      1;
   5:      24,     -50,       35,    -10,       1;
   6:    -120,     274,     -225,     85,     -15,     1;
   7:     720,   -1764,     1624,   -735,     175,   -21,     1;
   8:   -5040,   13068,   -13132,   6769,   -1960,   322,   -28,   1;
   9:   40320, -109584,   118124, -67284,   22449, -4536,   546, -36,   1;
  10: -362880, 1026576, -1172700, 723680, -269325, 63273, -9450, 870, -45, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1-(n-1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 - (n - 1)*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 - (n - 1)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A008276: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,  -1;
   3: 1,  -3,   2;
   4: 1,  -6,  11,    -6;
   5: 1, -10,  35,   -50,    24;
   6: 1, -15,  85,  -225,   274,    -120;
   7: 1, -21, 175,  -735,  1624,   -1764,    720;
   8: 1, -28, 322, -1960,  6769,  -13132,  13068,    -5040;
   9: 1, -36, 546, -4536, 22449,  -67284, 118124,  -109584,   40320;
  10: 1, -45, 870, -9450, 63273, -269325, 723680, -1172700, 1026576, -362880;
%F Formula
The row polynomials satisfy P(n,x) = (1-(n-1)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 - (n - 1)*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 - (n - 1)*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-68dd2f09e1ded893
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-68dd2f09e1ded893

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

Used in 2 proof steps

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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