A091042 → 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:
   1:      1;
   2:      4,      12;
   3:     16,      80,       80;
   4:     64,     448,      896,       448;
   5:    256,    2304,     6912,      7680,      2304;
   6:   1024,   11264,    45056,     78848,     56320,     11264;
   7:   4096,   53248,   266240,    638976,    745472,    372736,     53248;
   8:  16384,  245760,  1474560,   4505600,   7372800,   6193152,   2293760,    245760;
   9:  65536, 1114112,  7798784,  28966912,  61276160,  73531392,  46792704,  13369344,  1114112;
  10: 262144, 4980736, 39845888, 174325760, 453246976, 712245248, 657457152, 328728576, 74711040, 4980736;

The row polynomials satisfy P(n,x) = -16*x^2*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 12*x.

P[1, x_] := 1;
P[2, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[-16*x^2*P[n - 2, x] + (4 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 12*x; for(n = 3, N, my(i = n); v[i] = -16*x^2*v[i - 2] + (4 + 8*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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-8dd7014bf6954b08
• Coefficients of one sign. Every nonzero row, n >= 1: 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-8dd7014bf6954b08
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-8dd7014bf6954b08
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-8dd7014bf6954b08

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-8dd7014bf6954b08

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-1c0df90858517049 root: theorem check

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

Related sequences ?

Same fingerprint (4)

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

• A111125 → reverse
• A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
• A082985: Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
• A082985 → reverse

References

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