Real-rooted atlas

A204420: Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.

A204420 nonnegative coefficients only rational roots

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

Recurrence from OEIS A204420
\(P_{n}(x) = \left(2 - x - 6\,n + 2\,n x + 4\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20639
301209015225
4050404620126010511025
5036288037800013230018900945893025
60399168004557168018711000353430031185010395108056025
7062270208007628100480351134784079459380094594500567567013513518261468225

Show all 15 stored rows

Generated b-files: A001818 (row sums), A204420 ?

Computed in your browser ?

Technical details
Node id
n-1336d058dca9979e · revision 1 · JSON
Definition source
oeis_catalog:A204420
Fingerprint
f6-181e40cb14f29d52

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,               6,               3;
  3: 0,             120,              90,              15;
  4: 0,            5040,            4620,            1260,            105;
  5: 0,          362880,          378000,          132300,          18900,            945;
  6: 0,        39916800,        45571680,        18711000,        3534300,         311850,         10395;
  7: 0,      6227020800,      7628100480,      3511347840,      794593800,       94594500,       5675670,      135135;
  8: 0,   1307674368000,   1695306412800,    851804553600,   219534915600,    31783752000,    2610808200,   113513400,    2027025;
  9: 0, 355687428096000, 483353808537600, 260511047596800, 74194174454400, 12377274109200, 1250463614400, 75259384200, 2481078600, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^2-6*n+2+(2*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^2 - 6*n + 2 + (2*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^2 - 6*n + 2 + (2*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.

A204420: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0,               1;
  2: 0,               6,               3;
  3: 0,             120,              90,              15;
  4: 0,            5040,            4620,            1260,            105;
  5: 0,          362880,          378000,          132300,          18900,            945;
  6: 0,        39916800,        45571680,        18711000,        3534300,         311850,         10395;
  7: 0,      6227020800,      7628100480,      3511347840,      794593800,       94594500,       5675670,      135135;
  8: 0,   1307674368000,   1695306412800,    851804553600,   219534915600,    31783752000,    2610808200,   113513400,    2027025;
  9: 0, 355687428096000, 483353808537600, 260511047596800, 74194174454400, 12377274109200, 1250463614400, 75259384200, 2481078600, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^2-6*n+2+(2*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^2 - 6*n + 2 + (2*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^2 - 6*n + 2 + (2*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-8239b1c6b4764045
  • 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-8239b1c6b4764045
  • 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-8239b1c6b4764045

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 5 proof steps

Same fingerprint (2)

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