Real-rooted atlas

A225470: Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

A225470 nonnegative coefficients only rational roots

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

Recurrence from OEIS A225470
\(P_{n}(x) = \left(-1 + x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
2107118
38066151162
48808062312611944
51232012164404059540129160
620944021910880844141551275571524880
741888004591600183598836394439655241577111022480

Show all 15 stored rows

Generated b-files: A032031 (row sums), A225470 ?

Computed in your browser ?

Technical details
Node id
n-8229dbdd6a4b5cd9 · revision 2 · JSON
Definition source
oeis_catalog:A225470
Fingerprint
f6-217b33695a3e4949

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:          2,          1;
  2:         10,          7,          1;
  3:         80,         66,         15,         1;
  4:        880,        806,        231,        26,        1;
  5:      12320,      12164,       4040,       595,       40,       1;
  6:     209440,     219108,      80844,     14155,     1275,      57,      1;
  7:    4188800,    4591600,    1835988,    363944,    39655,    2415,     77,    1;
  8:   96342400,  109795600,   46819324,  10206700,  1276009,   95200,   4186,  100,   1;
  9: 2504902400, 2951028000, 1327098024, 312193524, 43382934, 3751209, 204036, 6786, 126, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*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[(3*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] = (3*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.

A225470: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:          2,          1;
  2:         10,          7,          1;
  3:         80,         66,         15,         1;
  4:        880,        806,        231,        26,        1;
  5:      12320,      12164,       4040,       595,       40,       1;
  6:     209440,     219108,      80844,     14155,     1275,      57,      1;
  7:    4188800,    4591600,    1835988,    363944,    39655,    2415,     77,    1;
  8:   96342400,  109795600,   46819324,  10206700,  1276009,   95200,   4186,  100,   1;
  9: 2504902400, 2951028000, 1327098024, 312193524, 43382934, 3751209, 204036, 6786, 126, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*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[(3*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] = (3*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-9b3c7c87e4baebfb
  • 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-8562aab4e97e2e3f, e-9b3c7c87e4baebfb, e-c0733b4d3f559bd0
  • 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-8562aab4e97e2e3f, e-9b3c7c87e4baebfb, e-c0733b4d3f559bd0
  • 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-9b3c7c87e4baebfb
  • 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-8562aab4e97e2e3f, e-c0733b4d3f559bd0
  • 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 = 3, 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-c4f0d53850ef9f92

Proof chains

4 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 7 proof steps

No candidate equalities or related sequences.

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. [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
  3. [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