A396532 First column of the triangular array with T(0, m) = binomial(2*m, m) and T(n, m) = T(n - 1, m + 3) - T(n - 1, m + 2).

1, 14, 490, 19382, 809354, 34841364, 1529402986, 68052212550, 3058446141450, 138511772397068, 6311106429976540, 288974736057466550, 13285601763128058890, 612898931850553795916, 28357246654163813673210, 1315325027404020086391882

Offset

0,2

Comments

The rows are generated by iterating the operator E^2*Delta, where E is the forward shift operator, E(f(m)) = f(m + 1), and Delta = E - 1 is the forward difference operator. Equivalently, E^2*Delta = E^2*(E - 1) = E^3 - E^2.

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = Sum_{j=0..n} (-1)^(n - j)*binomial(n, j)*binomial(2*(2*n + j), 2*n + j).

a(n) = (1/Pi)*Integral_{x = 0..4} x^(2*n)*(x - 1)^n/sqrt(x*(4 - x)) dx.

Example

Array begins:
  n = 0: 1       2        6         20       70      252      924
  n = 1: 14      50       182       672      2508    9438
  n = 2: 490     1826     6832      25740    97724
  n = 3: 19382   73484    280314    1076288
  n = 4: 809354  3108624  12022020

Crossrefs

Cf. A000984.

Sequence in context: A275348 A317635 A128051 * A217337 A251867 A240411

Adjacent sequences: A396528 A396530 A396531 * A396533 A396534 A396535

Keyword

nonn

Author

Dalton Heilig, May 28 2026

Status

approved