cp's OEIS Frontend

A273170 Denominators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2*m*x).

%I A273170 #11 Nov 18 2024 22:31:50
%S A273170 1,2,4,8,4,32,16,64,64,192,128,32,128,96,1024,256,512,256,1024,2048,
%T A273170 5120,1024,512,8192,3072,8192,5120,24576,2048,16384,16384,49152,16384,
%U A273170 81920,49152,114688,32768,4096,16384,4096,65536,4096,16384,28672,524288,65536,131072,32768,65536,131072,65536,32768,1835008,1048576,2359296,262144,131072,1048576,65536,524288,327680,2097152,1835008,4194304,2359296,10485760
%N A273170 Denominators of coefficient triangle for integrated even powers of cos(x) in terms of x and sin(2*m*x).
%C A273170 See the numerator triangle A273169, also for the formula of int(cos^(2*n)(x), x) in terms of x and sin(2*m*x).
%F A273170 a(n, m) = denominator(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 given in A273169 (where the sin arguments are falling).
%e A273170 See A273169, also for the rationals R(n,m).
%e A273170 The triangle a(n, m) begins:
%e A273170 n\m    0   1    2    3    4    5     6 ...
%e A273170 0:     1
%e A273170 1:     2   4
%e A273170 2:     8   4   32
%e A273170 3:    16  64   64  192
%e A273170 4:   128  32  128   96 1024
%e A273170 5:   256 512  256 1024 2048 5120
%e A273170 6:  1024 512 8192 3072 8192 5120 24576
%e A273170 ...
%e A273170 row 7: 2048 16384 16384 49152 16384 81920 49152 114688,
%e A273170 row 8: 32768 4096 16384 4096 65536 4096 16384 28672 524288,
%e A273170 row 9: 65536 131072 32768 65536 131072 65536 32768 1835008 1048576 2359296,
%e A273170 row 10: 262144 131072 1048576 65536 524288 327680 2097152 1835008 4194304 2359296 10485760,
%e A273170 ...
%o A273170 (PARI) a(n, m) = if (m == 0, denominator((1/2^(2*n))*binomial(2*n,n)), denominator((1/2^(2*n))*binomial(2*n, n-m)/m));
%o A273170 tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ _Michel Marcus_, Jun 19 2016
%Y A273170 Cf. A273169.
%K A273170 nonn,tabl,frac,easy
%O A273170 0,2
%A A273170 _Wolfdieter Lang_, Jun 13 2016