A079638 → 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:          4,          4;
  2:         24,         44,          16;
  3:        192,        488,         336,          64;
  4:       1920,       6056,        5936,        2176,         256;
  5:      23040,      85056,      105760,       56000,       12800,       1024;
  6:     322560,    1344864,     1993504,     1356480,      451840,      70656,       4096;
  7:    5160960,   23732544,    40352704,    33112576,    14291200,    3275776,     372736,     16384;
  8:   92897280,  463332672,   880475328,   837299456,   439124224,  131145728,   21962752,   1900544,   65536;
  9: 1857945600, 9927417600, 20695850496, 22198075136, 13586743296, 4956865536, 1085865984, 138706944, 9437184, 262144;

The row polynomials satisfy P(n,x) = -(2*n^2-2*n)*P(n-2,x) + (3*n+1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.

P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 2*n)*P[n - 2, x] + (3*n + 1 + 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; v[2] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 2*n)*v[i - 2] + (3*n + 1 + 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-3b42eaadd478c5dc
• 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-3b42eaadd478c5dc
• 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-3b42eaadd478c5dc
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-3b42eaadd478c5dc

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-3b42eaadd478c5dc

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-84c88fc8b81d60d6 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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023