A204420 → basis A013609

OEIS matches ?

• A079484: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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.

Triangle begins:
  0:                1;
  1:                1,                2;
  2:                9,               24,               12;
  3:              225,              690,              540,              120;
  4:            11025,            36960,            36120,            13440,             1680;
  5:           893025,          3192210,          3591000,          1738800,           378000,           30240;
  6:        108056025,        405903960,        504739620,        289396800,         83991600,        11975040,          665280;
  7:      18261468225,      71407225890,      95854498740,      62031289320,      21718897200,      4207563360,       423783360,       17297280;
  8:    4108830350625,   16614469872000,   23709478993200,   16832675059200,    6747690549600,   1598268672000,    221578156800,    16605388800,    518918400;
  9: 1187451971330625, 4941282024929250, 7416931404682800, 5670765377877600, 2522393520847200, 691321124894400, 118377222163200, 12332614694400, 714550636800, 17643225600;

The row polynomials satisfy P(n,x) = (4*n^2-4*n+1+(4*n-2)*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 + (4*n - 2)*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 + (4*n - 2)*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-2613ba1fbc2604f1
• 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-2613ba1fbc2604f1
• 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-2613ba1fbc2604f1

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 A013609 e-2613ba1fbc2604f1

  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 ?

Same fingerprint (2)

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

• A290319 → basis A013611
• A161198: Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).