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 A273171 #15 Dec 11 2019 07:38:10
%S A273171 1,3,1,5,5,1,35,7,7,1,63,7,9,9,1,231,55,33,55,11,1,429,429,143,143,13,
%T A273171 13,1,6435,5005,3003,195,455,105,15,1,12155,2431,1547,221,595,85,17,
%U A273171 17,1,46189,12597,12597,969,323,969,969,57,19,1,88179,146965,20349,14535,2261,20349,5985,133,105,21,1
%N A273171 Triangle for numerators of coefficients for integrated odd powers of cos(x) in terms sin((2*m+1)*x).
%C A273171 The triangle for the denominators is given in A273172.
%C A273171 Int(cos^(2*n+1)(x), x) = Sum_{m = 0..n} R(n, m)*sin((2*m+1)*x), n >= 0, with the rational triangle a(n, m)/A273172(n, m).
%C A273171 For the rational triangle for the odd powers of cos see A244420/A244421. See also the odd-indexed rows of A273496.
%C A273171 The signed rational triangle S(n, m) = R(n, m)*(-1)^(m+1) appears in the formula
%C A273171 Int(sin^(2*n+1)(x), x) = Sum_{m = 0..n} S(n, m)*cos((2*m+1)*x), n >= 0,
%D A273171 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 4.
%F A273171 a(n, m) = numerator(R(n, m)) with the rationals R(n, m) = (1/2^(2*n)) * binomial(2*n+1, n-m)/(2*m+1) for m = 0, ..., n, n >= 0. See the Gradstein-Ryshik reference (where the sin arguments are falling).
%e A273171 The triangle a(n, m) begins:
%e A273171 n\m 0 1 2 3 4 5 6 7 8 9
%e A273171 0: 1
%e A273171 1: 3 1
%e A273171 2: 5 5 1
%e A273171 3: 35 7 7 1
%e A273171 4: 63 7 9 9 1
%e A273171 5: 231 55 33 55 11 1
%e A273171 6: 429 429 143 143 13 13 1
%e A273171 7: 6435 5005 3003 195 455 105 15 1
%e A273171 8: 12155 2431 1547 221 595 85 17 17 1
%e A273171 9: 46189 12597 12597 969 323 969 969 57 19 1
%e A273171 ...
%e A273171 row 10: 88179 146965 20349 14535 2261 20349 5985 133 105 21 1,
%e A273171 ...
%e A273171 The rational triangle R(n, m) begins:
%e A273171 n\m 0 1 2 3 4 ...
%e A273171 0: 1/1
%e A273171 1: 3/4 1/12
%e A273171 2: 5/8 5/48 1/80
%e A273171 3: 35/64 7/64 7/320 1/448
%e A273171 4: 63/128 7/64 9/320 9/1792 1/2304
%e A273171 ...
%e A273171 row 5: 231/512 55/512 33/1024 55/7168 11/9216 1/11264,
%e A273171 row 6: 429/1024 429/4096 143/4096 143/14336 13/6144 13/45056 1/53248,
%e A273171 row 7: 6435/16384 5005/49152 3003/81920 195/16384 455/147456 105/180224 15/212992 1/245760,
%e A273171 row 8: 12155/32768 2431/24576 1547/40960 221/16384 595/147456 85/90112 17/106496 17/983040 1/1114112,
%e A273171 row 9: 46189/131072 12597/131072 12597/327680 969/65536 323/65536 969/720896 969/3407872 57/1310720 19/4456448 1/4980736,
%e A273171 row 10: 88179/262144 146965/1572864 20349/524288 14535/917504 2261/393216 20349/11534336 5985/13631488 133/1572864 105/8912896 21/19922944 1/22020096.
%e A273171 ...
%e A273171 n = 3: Int(cos^7(x), x) = (35/64)*sin(x) + (7/64)*sin(3*x) + (7/320)*sin(5*x) + (1/448)*sin(7*x). Gradstein-Rhyshik, p. 169, 2.513 16.
%e A273171 Int(sin^7(x), x) = -(35/64)*cos(x) + (7/64)*cos(3*x) - (7/320)*cos(5*x) + (1/448)*cos(7*x). Gradstein-Rhyshik, p. 169, 2.513 10.
%t A273171 T[MaxN_] := Function[{n}, With[{exp = Expand[(1/2)^(2 n + 1) (Exp[I x] + Exp[-I x])^(2 n + 1)]}, 2/(2 # + 1) Coefficient[exp, Exp[I (2 # + 1) x]] & /@ Range[0, n]]][#] & /@ Range[0, MaxN];
%t A273171 T[5] // ColumnForm (* _Bradley Klee_, Jun 14 2016 *)
%o A273171 (PARI) a(n, m) = numerator((1/2^(2*n))*binomial(2*n+1, n-m)/(2*m+1));
%o A273171 tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ _Michel Marcus_, Jun 19 2016
%Y A273171 Cf. A273172, A244420, A244421, A273496.
%K A273171 nonn,tabl,frac,easy
%O A273171 0,2
%A A273171 _Wolfdieter Lang_, Jun 13 2016