A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(x/(1-x))/(1-x)^k.

1, 1, 1, 1, 2, 3, 1, 3, 7, 13, 1, 4, 13, 34, 73, 1, 5, 21, 73, 209, 501, 1, 6, 31, 136, 501, 1546, 4051, 1, 7, 43, 229, 1045, 4051, 13327, 37633, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 394353, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 4596553

Offset

0,5

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} (j+k)*A(n-j,k)/(n-j)! for n > 0.

A(0,k) = 1, A(1,k) = k+1 and A(n,k) = (2*n-1+k)*A(n-1,k) - (n-1)*(n-2+k)*A(n-2,k) for n > 1.

Example

Square array begins:
     1,    1,    1,    1,     1, ... A000012;
     1,    2,    3,    4,     5, ... A000027;
     3,    7,   13,   21,    31, ... A002061;
    13,   34,   73,  136,   229, ... A135859;
    73,  209,  501, 1045,  1961, ...
   501, 1546, 4051, 9276, 19081, ...
Antidiagonal rows begin as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7, 13;
  1, 4, 13, 34,  73;
1, 5, 21, 73, 209, 501; - G. C. Greubel, Mar 09 2021

Mathematica

t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j, k]/(n-j)!, {j, n}]];
T[n_, k_]:= t[k, n-k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

Prog

(Sage)
@CachedFunction
def t(n, k): return 1 if n==0 else factorial(n-1)*sum( (j+k)*t(n-j, k)/factorial(n-j) for j in (1..n) )
def T(n, k): return t(k, n-k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021
(Magma)
function t(n, k)
  if n eq 0 then return 1;
  else return Factorial(n-1)*(&+[(j+k)*t(n-j, k)/Factorial(n-j): j in [1..n]]);
  end if; return t;
end function;
[t(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

Crossrefs

Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.

Main diagonal gives A152059.

Similar table: A086885, A088699, A176120.

Sequence in context: A069269 A193092 A263484 * A100324 A121424 A214978

Adjacent sequences: A293982 A293983 A293984 * A293986 A293987 A293988

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 21 2017

Status

approved