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%I #40 May 07 2018 22:25:42
%S 1,1,1,1,3,5,1,8,57,61,2,16,352,2763,1385,2,30,1280,38528,250737,
%T 50521,1,46,3522,249856,7869952,36581523,2702765,2,64,7970,1066590,
%U 90767360,2583554048,7828053417,199360981,2,96,15872,3487246,604935042,52975108096
%N Shanks's array c_{a,n} (a >= 1, n >= 0) that generalizes Euler and class numbers, read by antidiagonals upwards.
%H Lars Blomberg, <a href="/A235605/b235605.txt">Table of n, a(n) for n = 0..5049</a>
%H 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 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 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 A000364: 1, 1, 5, 61, 1385, 50521, 2702765,..
%e A000281: 1, 3, 57, 2763, 250737, 36581523, 7828053417,..
%e A000436: 1, 8, 352, 38528, 7869952, 2583554048, 1243925143552,..
%e A000490: 1,16, 1280, 249856, 90767360, 52975108096, 45344872202240,..
%e A000187: 2,30, 3522,1066590, 604935042, 551609685150, 737740947722562,..
%e A000192: 2,46, 7970,3487246, 2849229890, 3741386059246, 7205584123783010,..
%e A064068: 1,64,15872,9493504,10562158592,18878667833344,49488442978598912,..
%e ...
%t amax = 10; nmax = amax-1; km0 = 10; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[-a, 2k+1]/(2k+1)^s, {k, 0, km}]; c[1, n_, km_] := 2(2n)! L[1, 2n+1, km] (2/Pi)^(2n+1) // Round; c[a_ /; a>1, n_, km_] := (2n)! L[a, 2n+1, km] (2a/Pi)^(2n+1)/Sqrt[a] // Round; cc[km_] := cc[km] = Table[ c[a, n, km], {a, 1, amax}, {n, 0, nmax}]; cc[km0]; cc[km = 2km0]; While[ cc[km] != cc[km/2, km = 2km]]; A235605[a_, n_] := cc[km][[a, n+1 ]]; Table[ A235605[ a-n, n], {a, 1, amax}, {n, 0, a-1}] // Flatten (* _Jean-François Alcover_, Feb 05 2016 *)
%Y Rows: A000364 (Euler numbers), A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070, A064071, ...
%Y Columns: A000003 (class numbers), A000233, A000362, A000508, ...
%Y Cf. A235606.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Jan 22 2014
%E a(27) removed, a(29)-a(42) added, and typo in name corrected by _Lars Blomberg_, Sep 10 2015
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