A363365 Array read by ascending antidiagonals: A(1, k) = k; for n > 1, A(n, k) = (k + 1)*A(n-1, k) + k + 1 - n, with k > 0.

1, 2, 2, 3, 7, 3, 4, 21, 14, 4, 5, 62, 57, 23, 5, 6, 184, 228, 117, 34, 6, 7, 549, 911, 586, 207, 47, 7, 8, 1643, 3642, 2930, 1244, 333, 62, 8, 9, 4924, 14565, 14649, 7465, 2334, 501, 79, 9, 10, 14766, 58256, 73243, 44790, 16340, 4012, 717, 98, 10

Offset

1,2

Links

Table of n, a(n) for n=1..55.

Formula

A(n, k) = ((k - 1)*(k + 1)^(n+1) + k*n - k^2 + 1)/k^2.

O.g.f. of k-th column: x*(k - (k + 1)*x)/((1 - x)^2*(1 - (k + 1)*x)).

E.g.f. of k-th column: exp(x)*((k^2 - 1)*(exp(k*x) - 1) + k*x)/k^2.

A(2, n) = A008865(n+1).

Example

The array begins:
  1,   2,    3,     4,     5, ...
  2,   7,   14,    23,    34, ...
  3,  21,   57,   117,   207, ...
  4,  62,  228,   586,  1244, ...
  5, 184,  911,  2930,  7465, ...
  6, 549, 3642, 14649, 44790, ...
  ...

Mathematica

A[n_, k_]:=((k-1)*(k+1)^(n+1)+k*n-k^2+1)/k^2; Table[A[n-k+1, k], {n, 10}, {k, n}]//Flatten (* or *)
A[n_, k_]:=SeriesCoefficient[x*(k-(k+1)*x)/((1-x)^2*(1-(k+1)*x)), {x, 0, n}]; Table[A[n-k+1, k], {n, 10}, {k, n}]//Flatten (* or *)
A[n_, k_]:=n!SeriesCoefficient[Exp[x]((k^2-1)(Exp[k x]-1)+k x)/k^2, {x, 0, n}]; Table[A[n-k+1, k], {n, 10}, {k, n}]//Flatten

Crossrefs

Cf. A000027 (n=1 or k=1), A008865, A051846 (diagonal), A064017 (k=9), A353094 (k=2), A353095 (k=3), A353096 (k=4), A353097 (k=5), A353098 (k=6), A353099 (k=7), A353100 (k=8), A363366 (antidiagonal sums).

Sequence in context: A296019 A134232 A123934 * A208151 A203362 A368219

Adjacent sequences: A363362 A363363 A363364 * A363366 A363367 A363368

Keyword

nonn,tabl

Author

Stefano Spezia, May 29 2023

Status

approved