A069466 → basis A013611

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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,           8;
  2:         36,         144,           96;
  3:        400,        2400,         3840,          1280;
  4:       4900,       39200,       100800,         89600,         17920;
  5:      63504,      635040,      2257920,       3386880,       1935360,        258048;
  6:     853776,    10245312,     46569600,      99348480,      99348480,      39739392,       3784704;
  7:   11778624,   164900736,    913296384,    2536934400,    3690086400,    2656862208,     787218432,      56229888;
  8:  165636900,  2650190400,  17314577280,   59364264960,  114162048000,  121772851200,   66421555200,   15182069760,    843448320;
  9: 2363904400, 42550279200, 320378572800, 1308212505600, 3139710013440, 4485300019200, 3680246169600, 1577248358400, 286772428800, 12745441280;

The row polynomials satisfy n^2*P(n,x) = -(256*n^2-512*n+192)*x^2*P(n-2,x) + (16*n^2-16*n+4+(32*n^2-32*n+8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.

P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(256*n^2 - 512*n + 192)*x^2*P[n - 2, x] + (16*n^2 - 16*n + 4 + (32*n^2 - 32*n + 8)*x)*P[n - 1, x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(256*n^2 - 512*n + 192)*x^2*v[i - 2] + (16*n^2 - 16*n + 4 + (32*n^2 - 32*n + 8)*x)*v[i - 1])/(n^2)); 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-621bc4ded5b72353
• 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-621bc4ded5b72353
• 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-621bc4ded5b72353
• 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-621bc4ded5b72353

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-621bc4ded5b72353

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly 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-7dc194e46d1f26fe root: theorem check

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

Related sequences ?

Same fingerprint (1)

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

• A008459 → basis A013611

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