A204420 → basis A013611

OEIS matches ?

No OEIS matches.

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.

Triangle begins:
  0:                1;
  1:                1,                4;
  2:                9,               48,                48;
  3:              225,             1380,              2160,               960;
  4:            11025,            73920,            144480,            107520,             26880;
  5:           893025,          6384420,          14364000,          13910400,           6048000,            967680;
  6:        108056025,        811807920,        2018958480,        2315174400,        1343865600,         383201280,         42577920;
  7:      18261468225,     142814451780,      383417994960,      496250314560,      347502355200,      134642027520,      27122135040,       2214051840;
  8:    4108830350625,   33228939744000,    94837915972800,   134661400473600,   107963048793600,    51144597504000,   14181002035200,    2125489766400,    132843110400;
  9: 1187451971330625, 9882564049858500, 29667725618731200, 45366123023020800, 40358296333555200, 22122275996620800, 7576142218444800, 1578574680883200, 182924963020800, 9033331507200;

The row polynomials satisfy P(n,x) = (4*n^2-4*n+1+(8*n-4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n^2 - 4*n + 1 + (8*n - 4)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4*n^2 - 4*n + 1 + (8*n - 4)*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)

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-413dde327980a923
• 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-413dde327980a923
• 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-413dde327980a923

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

• basis_map A013611 e-413dde327980a923

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-8239b1c6b4764045 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.