Real-rooted atlas

A225471: Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

A225471 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 A225471
\(P_{n}(x) = \left(-1 + x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
22110132
3231131211384
4346521964463616144
56583545189106701130551122880
6151420511051822905993666023957812949120
74088353531354119895135512804191013254501105182575360

Show all 15 stored rows

Generated b-files: A047053 (row sums), A225471 ?

Computed in your browser ?

Technical details
Node id
n-9c5c208df8605708 · revision 1 · JSON
Definition source
oeis_catalog:A225471
Fingerprint
f6-83c2ad05f3223159

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:           3,           1;
  2:          21,          10,           1;
  3:         231,         131,          21,          1;
  4:        3465,        2196,         446,         36,         1;
  5:       65835,       45189,       10670,       1130,        55,        1;
  6:     1514205,     1105182,      290599,      36660,      2395,       78,      1;
  7:    40883535,    31354119,     8951355,    1280419,    101325,     4501,    105,     1;
  8:  1267389585,  1012861224,   308846124,   48644344,   4421494,   240856,   7756,   136,   1;
  9: 44358635475, 36717532425, 11822475564, 2011398164, 203396634, 12851454, 512316, 12516, 171, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4*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[(4*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] = (4*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.

A225471: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:           1;
  1:           3,           1;
  2:          21,          10,           1;
  3:         231,         131,          21,          1;
  4:        3465,        2196,         446,         36,         1;
  5:       65835,       45189,       10670,       1130,        55,        1;
  6:     1514205,     1105182,      290599,      36660,      2395,       78,      1;
  7:    40883535,    31354119,     8951355,    1280419,    101325,     4501,    105,     1;
  8:  1267389585,  1012861224,   308846124,   48644344,   4421494,   240856,   7756,   136,   1;
  9: 44358635475, 36717532425, 11822475564, 2011398164, 203396634, 12851454, 512316, 12516, 171, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4*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[(4*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] = (4*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-dad3570ee638379f
  • 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-381fad74fe44e78a, e-dad3570ee638379f
  • 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-381fad74fe44e78a, e-dad3570ee638379f
  • 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-dad3570ee638379f
  • 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-381fad74fe44e78a
  • 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 = 4, 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-836bdd982143692b

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 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