Real-rooted atlas

A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).

A094638 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A094638
\(P_{n}(x) = \left(1 + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21326
31611624
4110355024120
511585225274120720
6121175735162417647205040
7128322196067691313213068504040320

Show all 15 stored rows

Generated b-files: A094638, A130534 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-4d60c3dd8627c01d · revision 2 · JSON
Definition source
oeis_catalog:A094638
Fingerprint
f6-34eaf45657600f7a

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 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   2;
  3: 1,  6,  11,    6;
  4: 1, 10,  35,   50,    24;
  5: 1, 15,  85,  225,   274,    120;
  6: 1, 21, 175,  735,  1624,   1764,    720;
  7: 1, 28, 322, 1960,  6769,  13132,  13068,    5040;
  8: 1, 36, 546, 4536, 22449,  67284, 118124,  109584,   40320;
  9: 1, 45, 870, 9450, 63273, 269325, 723680, 1172700, 1026576, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (1+n*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[(1 + n*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {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] = (1 + 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)

Generated with polytool c48629374597.

A094638: triangle, offset 1, from the row P0
%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.

A130534: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      1,       1;
  2:      2,       3,       1;
  3:      6,      11,       6,      1;
  4:     24,      50,      35,     10,      1;
  5:    120,     274,     225,     85,     15,     1;
  6:    720,    1764,    1624,    735,    175,    21,    1;
  7:   5040,   13068,   13132,   6769,   1960,   322,   28,   1;
  8:  40320,  109584,  118124,  67284,  22449,  4536,  546,  36,  1;
  9: 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*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[(1 + n*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] = (1 + n*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 >= 0, is zero or real-rooted with all roots in (-inf, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-419a6747cb084155
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n 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-419a6747cb084155, e-986a227a8a459030, e-b29f35705824bef6
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-419a6747cb084155, e-986a227a8a459030, e-b29f35705824bef6
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-419a6747cb084155, e-b29f35705824bef6
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-986a227a8a459030, e-b29f35705824bef6

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 16 proof steps

Same fingerprint (6)

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