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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
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
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
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, May 29 2023
STATUS
approved