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A349271 Array A(n, k) that generalizes Euler numbers, class numbers, and tangent numbers, read by ascending antidiagonals.

%I A349271 #16 Oct 13 2023 06:49:04
%S A349271 1,1,1,1,1,1,1,2,3,2,2,4,8,11,5,2,4,16,46,57,16,1,6,30,128,352,361,61,
%T A349271 2,8,46,272,1280,3362,2763,272,2,8,64,522,3522,16384,38528,24611,1385,
%U A349271 2,12,96,904,7970,55744,249856,515086,250737,7936
%N A349271 Array A(n, k) that generalizes Euler numbers, class numbers, and tangent numbers, read by ascending antidiagonals.
%H A349271 William Y. C. Chen, Neil J. Y. Fan, and Jeffrey Y. T. Jia, <a href="http://dx.doi.org/10.1090/S0025-5718-2011-02520-2">The generating function for the Dirichlet series Lm(s)</a>, Mathematics of Computation, Vol. 81, No. 278, pp. 1005-1023, April 2012.
%H A349271 Ruth Lawrence and Don Zagier, <a href="https://doi.org/10.4310/AJM.1999.v3.n1.a5">Modular forms and quantum invariants of 3-manifolds</a>, Asian J. Math. 3 (1999), no. 1, 93-107.
%H A349271 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967) 689-694.
%H A349271 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-68-99652-X">Corrigendum: Generalized Euler and class numbers</a>, Math. Comp. 22, (1968) 699.
%H A349271 D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%e A349271 Seen as an array:
%e A349271 [1] 1,  1,   1,    2,     5,      16,       61,        272, ... [A000111]
%e A349271 [2] 1,  1,   3,   11,    57,     361,     2763,      24611, ... [A001586]
%e A349271 [3] 1,  2,   8,   46,   352,    3362,    38528,     515086, ... [A007289]
%e A349271 [4] 1,  4,  16,  128,  1280,   16384,   249856,    4456448, ... [A349264]
%e A349271 [5] 2,  4,  30,  272,  3522,   55744,  1066590,   23750912, ... [A349265]
%e A349271 [6] 2,  6,  46,  522,  7970,  152166,  3487246,   93241002, ... [A001587]
%e A349271 [7] 1,  8,  64,  904, 15872,  355688,  9493504,  296327464, ... [A349266]
%e A349271 [8] 2,  8,  96, 1408, 29184,  739328, 22634496,  806453248, ... [A349267]
%e A349271 [9] 2, 12, 126, 2160, 49410, 1415232, 48649086, 1951153920, ... [A349268]
%e A349271 .
%e A349271 Seen as a triangle:
%e A349271 [1] 1;
%e A349271 [2] 1, 1;
%e A349271 [3] 1, 1,  1;
%e A349271 [4] 1, 2,  3,   2;
%e A349271 [5] 2, 4,  8,  11,    5;
%e A349271 [6] 2, 4, 16,  46,   57,    16;
%e A349271 [7] 1, 6, 30, 128,  352,   361,    61;
%e A349271 [8] 2, 8, 46, 272, 1280,  3362,  2763,   272;
%e A349271 [9] 2, 8, 64, 522, 3522, 16384, 38528, 24611, 1385;
%Y A349271 A235605 (array generalized Euler secant numbers).
%Y A349271 A235606 (array generalized Euler tangent numbers).
%Y A349271 A349264 (overview generating functions).
%Y A349271 Rows: A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.
%Y A349271 Columns: A000003 (class numbers), A000061, A000233, A000176, A000362, A000488, A000508, A000518.
%Y A349271 Cf. A349263 (main diagonal).
%K A349271 nonn,tabl
%O A349271 1,8
%A A349271 _Peter Luschny_, Nov 23 2021