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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

%I #4 Dec 09 2018 11:40:05

%S 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,

%T 36,129,304,433,294,1,1,8,49,221,705,1519,2029,1309,1,1,9,64,349,1416,

%U 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

%N 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.

%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>

%F E.g.f. of column k: exp(x)*(sec(x) + tan(x))^k.

%e 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! + ...

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, ...

%e 1, 4, 9, 16, 25, 36, ...

%e 1, 9, 29, 67, 129, 221, ...

%e 1, 24, 105, 304, 705, 1416, ...

%e 1, 77, 433, 1519, 4145, 9601, ...

%t Table[Function[k, n! SeriesCoefficient[Exp[x] (Sec[x] + Tan[x])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

%Y Columns k=0..3 give A000012, A000667, A292756, A292759.

%Y Main diagonal gives A296793.

%Y Cf. A322267.

%K nonn,tabl

%O 0,5

%A _Ilya Gutkovskiy_, Dec 01 2018

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)