Real-rooted atlas

A132393: Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.

A132393 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A132393
\(P_{n}(x) = \left(-1 + x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20112
302316
406116124
50245035101120
6012027422585151720
70720176416247351752115040

Show all 15 stored rows

Generated b-files: A000142 (row sums), A094638 (reversed triangle), A130534, A132393 ?

Computed in your browser ?

Technical details
Node id
n-b9fbfb57d9269e2f · revision 1 · JSON
Definition source
oeis_catalog:A132393
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: 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[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] = (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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A094638: reversed triangle, offset 1, from the row P1, 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.

A130534: triangle, offset 0, from the row P1, low zeros dropped
%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 ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (n+x)*P(n-1,x) for n >= 1, with P(0,x) = x.
%t Mathematica
P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[(n + x)*P[n - 1, x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = (n + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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.

A132393: triangle, offset 0, from the row P0
%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.

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-edbf92ee2e446027
  • 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-edbf92ee2e446027
  • 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-edbf92ee2e446027
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: (-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-4392ab16ca9688ca

Proof chains

2 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 15 proof steps

Same fingerprint (6)

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

References

  1. [Bre89] Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989. doi:10.1090/memo/0413
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741