Real-rooted atlas

A048994: Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.

A028338 → x ↦ −2x−1 (shift -1)

A048994 only rational roots

Proved for all n: all roots real, in [−1/2, ∞) · interlacing (weak) what this means

Recurrence from OEIS A048994
\(P_{n}(x) = \left(2 + x - n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2011
30-110
402-310
50-611-610
6024-5035-1010
70-120274-22585-1510
80720-17641624-735175-2110
90-504013068-131326769-1960322-2810
10040320-109584118124-6728422449-4536546-3610
110-3628801026576-1172700723680-26932563273-9450870-4510
1203628800-1062864012753576-84095003416930-902055157773-181501320-5510
130-39916800120543840-150917976105258076-4599573013339535-2637558357423-326701925-6610
140479001600-14864428801931559552-1414014888657206836-20607015044990231-6926634749463-557702717-7810
150-622702080019802759040-2659671705620313753096-99577037563336118786-790943153135036473-166696531474473-910913731-9110

Generated b-files: A008275, A008276 (reversed triangle), A048994, A054654 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-a52b649d7348b7a5 · revision 1 · JSON
Definition source
oeis_catalog:A048994
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: 0,     1;
   3: 0,    -1,       1;
   4: 0,     2,      -3,      1;
   5: 0,    -6,      11,     -6,      1;
   6: 0,    24,     -50,     35,    -10,     1;
   7: 0,  -120,     274,   -225,     85,   -15,     1;
   8: 0,   720,   -1764,   1624,   -735,   175,   -21,   1;
   9: 0, -5040,   13068, -13132,   6769, -1960,   322, -28,   1;
  10: 0, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2-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[-(n - 2 - 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] = -(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, v[i])))};
T(10)

Generated with polytool c48629374597.

A008275: triangle, offset 1, from the row P2, low zeros dropped
%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 ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = -(n-1-x)*P(n-1,x) for n >= 2, with P(1,x) = x.
%t Mathematica
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[-(n - 1 - x)*P[n - 1, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = -(n - 1 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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.

A008276: reversed triangle, offset 1, from the row P2, low zeros dropped
%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
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = -(n-1-x)*P(n-1,x) for n >= 2, with P(1,x) = x.
%t Mathematica
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[-(n - 1 - x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = -(n - 1 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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.

A048994: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 0,     1;
  2: 0,    -1,       1;
  3: 0,     2,      -3,      1;
  4: 0,    -6,      11,     -6,      1;
  5: 0,    24,     -50,     35,    -10,     1;
  6: 0,  -120,     274,   -225,     85,   -15,     1;
  7: 0,   720,   -1764,   1624,   -735,   175,   -21,   1;
  8: 0, -5040,   13068, -13132,   6769, -1960,   322, -28,   1;
  9: 0, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-1-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n - 1 - x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(n - 1 - 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)

Generated with polytool c48629374597.

A054654: reversed triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,   0;
  2: 1,  -1,   0;
  3: 1,  -3,   2,     0;
  4: 1,  -6,  11,    -6,     0;
  5: 1, -10,  35,   -50,    24,      0;
  6: 1, -15,  85,  -225,   274,   -120,      0;
  7: 1, -21, 175,  -735,  1624,  -1764,    720,       0;
  8: 1, -28, 322, -1960,  6769, -13132,  13068,   -5040,     0;
  9: 1, -36, 546, -4536, 22449, -67284, 118124, -109584, 40320, 0;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(n-1-x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n - 1 - x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(n - 1 - x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1/2, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6e459812af5dc78f
  • 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-6e459812af5dc78f

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