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%I #20 Feb 05 2016 09:01:56
%S 1,1,2,2,11,16,4,46,361,272,4,128,3362,24611,7936,6,272,16384,515086,
%T 2873041,353792,8,522,55744,4456448,135274562,512343611,22368256,8,
%U 904,152166,23750912,2080374784,54276473326,129570724921,1903757312,12,1408,355688
%N Shanks's array d_{a,n} (a >= 1, n >= 1) that generalizes the tangent numbers, read by antidiagonals upwards.
%D D. Shanks. "Generalized Euler and Class Numbers." Math. Comput. 21, 689-694, 1967. Math. Comput. 22, 699, 1968.
%H Lars Blomberg, <a href="/A235606/b235606.txt">Table of n, a(n) for n = 1..5050</a>
%H 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]
%F Shanks gives recurrences.
%e The array begins:
%e A000182: 1, 2, 16, 272, 7936, 353792, ...
%e A000464: 1, 11, 361, 24611, 2873041, 512343611, ...
%e A000191: 2, 46, 3362, 515086, 135274562, 54276473326, ...
%e A000318: 4,128, 16384, 4456448, 2080374784, 1483911200768, ...
%e A000320: 4,272, 55744, 23750912, 17328937984, 19313964388352, ...
%e A000411: 6,522,152166, 93241002, 97949265606,157201459863882, ...
%e A064072: 8,904,355688,296327464,423645846728,925434038426824, ...
%e ...
%t amax = nmax = 10; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[ -a, 2k+1]/(2k+1)^s, {k, 0, km}]; d[1, n_, km_] := 2(2n-1)! L[-1, 2n, km] (2/Pi)^(2n) // Round; d[a_ /; a>1, n_, km_] := (2n-1)! L[-a, 2n, km] (2a/ Pi)^(2n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[a, n, km], {a, 1, amax}, {n, 1, nmax}]; dd[km0]; dd[km = 2km0]; While[dd[km] != dd[km/2, km = 2km]]; A235606 = dd[km]; Table[A235606[[ a-n+1, n]], {a, 1, amax}, {n, 1, a}] // Flatten (* _Jean-François Alcover_, Feb 05 2016 *)
%Y Rows: A000182 (tangent numbers), A000464, A000191, A000318, A000320, A000411, A064072-A064075, ...
%Y Columns: A000061, A000176, A000488, A000518, ...
%Y Cf. A235605.
%K nonn,tabl
%O 1,3
%A _N. J. A. Sloane_, Jan 22 2014
%E More terms from _Lars Blomberg_, Sep 07 2015
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