cp's OEIS Frontend

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.

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A273171 Triangle for numerators of coefficients for integrated odd powers of cos(x) in terms sin((2*m+1)*x).

Original entry on oeis.org

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, 13, 1, 6435, 5005, 3003, 195, 455, 105, 15, 1, 12155, 2431, 1547, 221, 595, 85, 17, 17, 1, 46189, 12597, 12597, 969, 323, 969, 969, 57, 19, 1, 88179, 146965, 20349, 14535, 2261, 20349, 5985, 133, 105, 21, 1
Offset: 0

Views

Author

Wolfdieter Lang, Jun 13 2016

Keywords

Comments

The triangle for the denominators is given in A273172.
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).
For the rational triangle for the odd powers of cos see A244420/A244421. See also the odd-indexed rows of A273496.
The signed rational triangle S(n, m) = R(n, m)*(-1)^(m+1) appears in the formula
Int(sin^(2*n+1)(x), x) = Sum_{m = 0..n} S(n, m)*cos((2*m+1)*x), n >= 0,

Examples

			The triangle a(n, m) begins:
n\m    0     1     2   3   4   5   6  7  8 9
0:     1
1:     3     1
2:     5     5     1
3:    35     7     7   1
4:    63     7     9   9  1
5:   231    55    33  55  11   1
6:   429   429   143 143  13  13   1
7:  6435  5005  3003 195 455 105  15  1
8: 12155  2431  1547 221 595  85  17 17  1
9: 46189 12597 12597 969 323 969 969 57 19 1
...
row 10: 88179 146965 20349 14535 2261 20349 5985 133 105 21 1,
...
The rational triangle R(n, m) begins:
n\m   0    1     2    3      4     ...
0:   1/1
1:   3/4  1/12
2:   5/8  5/48 1/80
3:  35/64 7/64 7/320 1/448
4: 63/128 7/64 9/320 9/1792 1/2304
...
row 5: 231/512 55/512 33/1024 55/7168 11/9216 1/11264,
row 6: 429/1024 429/4096 143/4096 143/14336 13/6144 13/45056 1/53248,
row 7: 6435/16384 5005/49152 3003/81920 195/16384 455/147456 105/180224 15/212992 1/245760,
row 8: 12155/32768 2431/24576 1547/40960 221/16384 595/147456 85/90112 17/106496 17/983040 1/1114112,
row 9: 46189/131072 12597/131072 12597/327680 969/65536 323/65536 969/720896 969/3407872 57/1310720 19/4456448 1/4980736,
row 10: 88179/262144 146965/1572864 20349/524288 14535/917504 2261/393216 20349/11534336 5985/13631488 133/1572864 105/8912896 21/19922944 1/22020096.
...
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.
  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.

References

  • 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.

Crossrefs

Cf. A273172, A244420, A244421, A273496.

Programs

  • Mathematica
    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[5] // ColumnForm (* Bradley Klee, Jun 14 2016 *)
  • PARI
    a(n, m) = numerator((1/2^(2*n))*binomial(2*n+1, n-m)/(2*m+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ Michel Marcus, Jun 19 2016

Formula

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).

A273169 Numerators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2*m*x).

Original entry on oeis.org

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, 1, 429, 3003, 1001, 1001, 91, 91, 7, 1, 6435, 715, 1001, 91, 455, 7, 5, 1, 1, 12155, 21879, 1989, 1547, 1071, 153, 17, 153, 9, 1, 46189, 20995, 62985, 1615, 4845, 969, 1615, 285, 95, 5, 1
Offset: 0

Views

Author

Wolfdieter Lang, Jun 13 2016

Keywords

Comments

The denominator triangle is given in A273170.
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).
For the rational triangle for the even powers of cos see A273167/A273168. See also the even-indexed rows of A273496.
For the integral over odd powers of cos see the rational triangle A273171/A273172.
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.

Examples

			The triangle a(n, m) begins:
n\m    0     1    2    3    4   5  6   7 8 9
0:     1
1:     1     1
2:     3     1    1
3:     5    15    3    1
4:    35     7    7    1    1
5:    63   105   15   15    5   1
6:   231    99  495   55   33   3  1
7:   429  3003 1001 1001   91  91  7   1
8:  6435   715 1001   91  455   7  5   1 1
9: 12155 21879 1989 1547 1071 153 17 153 9 1
...
row 10: 46189 20995 62985 1615 4845 969 1615 285 95 5 1,
...
The rational triangle R(n, m) begins:
n\m   0      1     2      3      4     ...
0:   1/1
1:   1/2    1/4
2:   3/8    1/4   1/32
3:   5/16  15/64  3/64   1/192
4:  35/128  7/32  7/128  1/96  1/1024
...
row 5: 63/256 105/512 15/256 15/1024 5/2048 1/5120,
row 6: 231/1024 99/512 495/8192 55/3072 33/8192 3/5120 1/24576,
row 7: 429/2048 3003/16384 1001/16384 1001/49152 91/16384 91/81920 7/49152 1/114688,
row 8: 6435/32768 715/4096 1001/16384 91/4096 455/65536 7/4096 5/16384 1/28672 1/524288,
row 9: 12155/65536 21879/131072 1989/32768 1547/65536 1071/131072 153/65536 17/32768 153/1835008 9/1048576 1/2359296,
row 10: 46189/262144 20995/131072 62985/1048576 1615/65536 4845/524288 969/327680 1615/2097152 285/1835008 95/4194304 5/2359296 1/10485760,
...
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).
  Int(sin^6(x), x) = (5/16)*x - (15/64)*sin(2*x) + (3/64)*sin(4*x) - (1/192)*sin(6*x).

References

  • 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.

Crossrefs

Cf. A273170, A273167, A273168, A273496, A273171, A273172.

Programs

  • Mathematica
    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[5] // ColumnForm (* Bradley Klee, Jun 14 2016 *)
  • 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));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ Michel Marcus, Jun 19 2016

Formula

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).
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