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 A273169 #22 Nov 18 2024 22:31:55
%S A273169 1,1,1,3,1,1,5,15,3,1,35,7,7,1,1,63,105,15,15,5,1,231,99,495,55,33,3,
%T A273169 1,429,3003,1001,1001,91,91,7,1,6435,715,1001,91,455,7,5,1,1,12155,
%U A273169 21879,1989,1547,1071,153,17,153,9,1,46189,20995,62985,1615,4845,969,1615,285,95,5,1
%N A273169 Numerators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2*m*x).
%C A273169 The denominator triangle is given in A273170.
%C A273169 Int(cos^(2*n)(x), x) = R(n, 0)*x + Sum_{m = 1..n} R(n, m)*sin(2*m*x), n >= 0, with the rational triangle a(n, m)/A273170(n, m).
%C A273169 For the rational triangle for the even powers of cos see A273167/A273168. See also the even-indexed rows of A273496.
%C A273169 For the integral over odd powers of cos see the rational triangle A273171/A273172.
%C A273169 The signed triangle S(n, m) = R(n, m)*(-1)^m appears in the integral of even powers of sin as Int(sin^(2*n)(x), x) = S(n , 0)*x + Sum_{m = 1..n} S(n, m)*sin(2*m*x), n >= 0.
%D A273169 I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981, pp. 168-169, 2.513 1. and 3.
%F A273169 a(n, m) = numerator(R(n, m)) with the rationals R(n, m) defined by R(n, 0) = (1/2^(2*n))*binomial(2*n,n) and R(n, m) = (1/2^(2*n))*binomial(2*n, n-m)/m for m = 1, ..., n, n >= 0. See the Gradstein-Ryshik reference (where the sin arguments are falling).
%e A273169 The triangle a(n, m) begins:
%e A273169 n\m 0 1 2 3 4 5 6 7 8 9
%e A273169 0: 1
%e A273169 1: 1 1
%e A273169 2: 3 1 1
%e A273169 3: 5 15 3 1
%e A273169 4: 35 7 7 1 1
%e A273169 5: 63 105 15 15 5 1
%e A273169 6: 231 99 495 55 33 3 1
%e A273169 7: 429 3003 1001 1001 91 91 7 1
%e A273169 8: 6435 715 1001 91 455 7 5 1 1
%e A273169 9: 12155 21879 1989 1547 1071 153 17 153 9 1
%e A273169 ...
%e A273169 row 10: 46189 20995 62985 1615 4845 969 1615 285 95 5 1,
%e A273169 ...
%e A273169 The rational triangle R(n, m) begins:
%e A273169 n\m 0 1 2 3 4 ...
%e A273169 0: 1/1
%e A273169 1: 1/2 1/4
%e A273169 2: 3/8 1/4 1/32
%e A273169 3: 5/16 15/64 3/64 1/192
%e A273169 4: 35/128 7/32 7/128 1/96 1/1024
%e A273169 ...
%e A273169 row 5: 63/256 105/512 15/256 15/1024 5/2048 1/5120,
%e A273169 row 6: 231/1024 99/512 495/8192 55/3072 33/8192 3/5120 1/24576,
%e A273169 row 7: 429/2048 3003/16384 1001/16384 1001/49152 91/16384 91/81920 7/49152 1/114688,
%e A273169 row 8: 6435/32768 715/4096 1001/16384 91/4096 455/65536 7/4096 5/16384 1/28672 1/524288,
%e A273169 row 9: 12155/65536 21879/131072 1989/32768 1547/65536 1071/131072 153/65536 17/32768 153/1835008 9/1048576 1/2359296,
%e A273169 row 10: 46189/262144 20995/131072 62985/1048576 1615/65536 4845/524288 969/327680 1615/2097152 285/1835008 95/4194304 5/2359296 1/10485760,
%e A273169 ...
%e A273169 n = 3: Int(cos^6(x), x) = (5/16)*x + (15/64)*sin(2*x) + (3/64)*sin(4*x) + (1/192)*sin(6*x).
%e A273169 Int(sin^6(x), x) = (5/16)*x - (15/64)*sin(2*x) + (3/64)*sin(4*x) - (1/192)*sin(6*x).
%t A273169 T[MaxN_] := Function[{n}, With[{exp = Expand[(1/2)^(2 n) (Exp[I x] + Exp[-I x])^(2 n)]}, Prepend[1/# Coefficient[exp, Exp[I 2 # x]] & /@ Range[1, n], exp /. {Exp[_] -> 0}]]][#] & /@ Range[0, MaxN];
%t A273169 T[5] // ColumnForm (* _Bradley Klee_, Jun 14 2016 *)
%o A273169 (PARI) a(n, m) = if (m == 0, numerator((1/2^(2*n))*binomial(2*n,n)), numerator((1/2^(2*n))*binomial(2*n, n-m)/m));
%o A273169 tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ _Michel Marcus_, Jun 19 2016
%Y A273169 Cf. A273170, A273167, A273168, A273496, A273171, A273172.
%K A273169 nonn,tabl,frac,easy
%O A273169 0,4
%A A273169 _Wolfdieter Lang_, Jun 13 2016