This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A235605 #51 Oct 25 2024 15:45:25
%S A235605 1,1,1,1,3,5,1,8,57,61,2,16,352,2763,1385,2,30,1280,38528,250737,
%T A235605 50521,1,46,3522,249856,7869952,36581523,2702765,2,64,7970,1066590,
%U A235605 90767360,2583554048,7828053417,199360981,2,96,15872,3487246,604935042,52975108096
%N A235605 Shanks's array c_{a,n} (a >= 1, n >= 0) that generalizes Euler and class numbers, read by antidiagonals upwards.
%H A235605 Matthew House, <a href="/A235605/b235605.txt">Table of n, a(n) for n = 1..10011</a> (first 141 antidiagonals; first 100 antidiagonals from Lars Blomberg)
%H A235605 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 A235605 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 A235605 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 A235605 Shanks gives recurrences.
%e A235605 The array begins:
%e A235605 A000364: 1, 1, 5, 61, 1385, 50521, 2702765,..
%e A235605 A000281: 1, 3, 57, 2763, 250737, 36581523, 7828053417,..
%e A235605 A000436: 1, 8, 352, 38528, 7869952, 2583554048, 1243925143552,..
%e A235605 A000490: 1,16, 1280, 249856, 90767360, 52975108096, 45344872202240,..
%e A235605 A000187: 2,30, 3522,1066590, 604935042, 551609685150, 737740947722562,..
%e A235605 A000192: 2,46, 7970,3487246, 2849229890, 3741386059246, 7205584123783010,..
%e A235605 A064068: 1,64,15872,9493504,10562158592,18878667833344,49488442978598912,..
%e A235605 ...
%t A235605 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 *)
%t A235605 ccs[b_, nm_] := With[{ns = Range[0, nm]}, (-1)^ns If[Mod[b, 4] == 3, Sum[JacobiSymbol[k, b] (b - 4 k)^(2 ns), {k, 1, (b - 1)/2}], Sum[JacobiSymbol[-b, 2 k + 1] (b - (2 k + 1))^(2 ns), {k, 0, (b - 2)/2}]]];
%t A235605 csfs[1, nm_] := csfs[1, nm] = (2 Range[0, nm])! CoefficientList[Series[Sec[x], {x, 0, 2 nm}], x^2];
%t A235605 csfs[b_, nm_] := csfs[b, nm] = Fold[Function[{cs, cc}, Append[cs, cc - Sum[cs[[-i]] (-b^2)^i Binomial[2 Length[cs], 2 i], {i, Length[cs]}]]], {}, ccs[b, nm]];
%t A235605 rowA235605[a_, nm_] := With[{facs = FactorInteger[a], ns = Range[0, nm]}, With[{b = Times @@ (#^Mod[#2, 2] &) @@@ facs}, If[a == b, csfs[b, nm], If[b == 1, 1/2, 1] csfs[b, nm] Sqrt[a/b]^(4 ns + 1) Times @@ Cases[facs, {p_, e_} /; p > 2 && e > 1 :> 1 - JacobiSymbol[-b, p]/p^(2 ns + 1)]]]];
%t A235605 arr = Table[rowA235605[a, 10], {a, 10}];
%t A235605 Flatten[Table[arr[[r - n + 1, n + 1]], {r, 0, Length[arr] - 1}, {n, 0, r}]] (* _Matthew House_, Sep 07 2024 *)
%Y A235605 Rows: A000364 (Euler numbers), A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070, A064071, ...
%Y A235605 Columns: A000003 (class numbers), A000233, A000362, A000508, ...
%Y A235605 Cf. A235606.
%K A235605 nonn,tabl
%O A235605 1,5
%A A235605 _N. J. A. Sloane_, Jan 22 2014
%E A235605 a(27) removed, a(29)-a(42) added, and typo in name corrected by _Lars Blomberg_, Sep 10 2015
%E A235605 Offset corrected by _Andrew Howroyd_, Oct 25 2024