A069466 → basis A007318

OEIS matches ?

• A268555: 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:          4,           2;
  2:         36,          36,           6;
  3:        400,         600,         240,          20;
  4:       4900,        9800,        6300,        1400,          70;
  5:      63504,      158760,      141120,       52920,        7560,        252;
  6:     853776,     2561328,     2910600,     1552320,      388080,      38808,       924;
  7:   11778624,    41225184,    57081024,    39639600,    14414400,    2594592,    192192,     3432;
  8:  165636900,   662547600,  1082161080,   927566640,   445945500,  118918800,  16216200,   926640,   12870;
  9: 2363904400, 10637569800, 20023660800, 20440820400, 12264492240, 4380175800, 898497600, 96267600, 4375800, 48620;

The row polynomials satisfy n^2*P(n,x) = -(16*n^2-32*n+12)*x^2*P(n-2,x) + (16*n^2-16*n+4+(8*n^2-8*n+2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n^2 - 32*n + 12)*x^2*P[n - 2, x] + (16*n^2 - 16*n + 4 + (8*n^2 - 8*n + 2)*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(16*n^2 - 32*n + 12)*x^2*v[i - 2] + (16*n^2 - 16*n + 4 + (8*n^2 - 8*n + 2)*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-2cd9cd27e716475b
• 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-2cd9cd27e716475b
• 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-2cd9cd27e716475b
• 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-2cd9cd27e716475b

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 A007318 e-2cd9cd27e716475b

  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 (4)

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

• A008459 → x ↦ x/(x+1)
• A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) …
• A008459 → x ↦ x+1
• A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.

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