A085841 → 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:         7,           4;
  2:        61,          88,           16;
  3:       547,        1596,          720,           64;
  4:      4921,       26352,        22240,         4864,          256;
  5:     44287,      413620,       578272,       228480,        29440,        1024;
  6:    398581,     6311368,     13676912,      8565504,      1954560,      165888,       4096;
  7:   3587227,    94754412,    305630512,    282476224,     99662080,    14787584,     888832,     16384;
  8:  32285041,  1409076960,   6588726464,   8598616576,   4356281856,   983506944,  102416384,   4587520,    65536;
  9: 290565367, 20835210340, 138718498496, 248203981056, 172467905024, 54940121088, 8608137216, 663552000, 23003136, 262144;

The row polynomials satisfy P(n,x) = -(9+40*x+16*x^2)*P(n-2,x) - (40*x+32*x^2)*d/dx P(n-2,x) - 16*x^2*d^2/dx^2 P(n-2,x) + (10+8*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.

P[0, x_] := 1;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9 + 40*x + 16*x^2)*P[n - 2, x] - (40*x + 32*x^2)*D[P[n - 2, x], x] - 16*x^2*D[P[n - 2, x], {x, 2}] + (10 + 8*x)*P[n - 1, x] + 8*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; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 40*x + 16*x^2)*v[i - 2] - (40*x + 32*x^2)*deriv(v[i - 2], x) - 16*x^2*deriv(deriv(v[i - 2], x), x) + (10 + 8*x)*v[i - 1] + 8*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-db4d4459bbf9061f
• 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-db4d4459bbf9061f
• 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-db4d4459bbf9061f

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-db4d4459bbf9061f

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

  • premise (sequence): Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-c0dcd56725ae4481 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [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
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023