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A322268
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x)*(sec(x) + tan(x))^k.
2
1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 29, 24, 1, 1, 6, 25, 67, 105, 77, 1, 1, 7, 36, 129, 304, 433, 294, 1, 1, 8, 49, 221, 705, 1519, 2029, 1309, 1, 1, 9, 64, 349, 1416, 4145, 8386, 10709, 6664, 1, 1, 10, 81, 519, 2569, 9601, 26385, 51007, 63025, 38177, 1, 1, 11, 100, 737, 4320, 19777, 69406, 181969, 340024, 409713, 243034, 1
OFFSET
0,5
FORMULA
E.g.f. of column k: exp(x)*(sec(x) + tan(x))^k.
EXAMPLE
E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k + 1)^2*x^2/2! + (k^3 + 3*k^2 + 4*k + 1)*x^3/3! + (k^4 + 4*k^3 + 10*k^2 + 8*k + 1)*x^4/4! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
1, 4, 9, 16, 25, 36, ...
1, 9, 29, 67, 129, 221, ...
1, 24, 105, 304, 705, 1416, ...
1, 77, 433, 1519, 4145, 9601, ...
MATHEMATICA
Table[Function[k, n! SeriesCoefficient[Exp[x] (Sec[x] + Tan[x])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
CROSSREFS
Columns k=0..3 give A000012, A000667, A292756, A292759.
Main diagonal gives A296793.
Cf. A322267.
Sequence in context: A071922 A138028 A009999 * A144823 A098446 A098447
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Dec 01 2018
STATUS
approved