A204420 → basis A008277

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,                1;
  2:                9,               15,                3;
  3:              225,              495,              180,               15;
  4:            11025,            29295,            14805,             2310,             105;
  5:           893025,          2735775,          1729350,           382725,           33075,            945;
  6:        108056025,        370945575,        277390575,         77962500,         9667350,         530145,         10395;
  7:      18261468225,      68916822975,      58913454600,      19822277475,      3149996850,      249729480,       9459450,       135135;
  8:    4108830350625,   16813959537375,   16056683267625,    6235716737250,   1195083264375,   122168796750,    6725668950,    186486300,    2027025;
  9: 1187451971330625, 5214921734397375, 5472146007578250, 2393107306214625, 532651909915125, 66007545478875, 4706606104200, 190422782550, 4031752725, 34459425;

The row polynomials satisfy P(n,x) = (4*n^2-4*n+1+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*d/dx 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 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], 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 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x)); 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-86d697d3c8220248
• 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-86d697d3c8220248
• 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-86d697d3c8220248

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 A008277 e-86d697d3c8220248

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • 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.

References

1. [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