Summation proof

[Albert Chan]

Here is HP71B code, that the coefficients are used.
It did work, but code really not suitable with approximate numbers.

10 DESTROY ALL @ OPTION BASE 0 @ INPUT "M= ";M
20 DIM C(M),D(M) @ N=M DIV 2 @ S=(-1)^(M-N-N) @ D(N)=0
30 FOR J=1 TO N @ D(N+J)=J^M @ D(N-J)=S*D(N+J) @ NEXT J
40 IF S<0 THEN D(M)=(N+1)^M
50 FOR I=0 TO M-1 @ FOR J=M TO I+1 STEP -1 @ D(J)=D(J)-D(J-1) @ NEXT J @ NEXT I
100 K=1 @ S=0 @ N1=N+1 @ C(N)=1/N1
110 FOR J=N TO 1 STEP -1 @ K=K*J/(N1-J) @ S=S+1/J @ C(N-J)=K*S @ NEXT J
120 FOR J=N1 TO M @ C(J)=C(J-1)*(N1-J-1)/(J+1) @ NEXT J
130 DISP "B=";DOT(C,D)

>run
M= 6
B= 2.38095210526E-2
>1/res
 42.0000048632

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The simple and accurate solution is with zeta correction.

>5 DEF FNB(M)=(-1)^(M/2+1)*(IP(2*FACT(M)/(PI^M)*(1+3^(-M)))+.5)/(2^M-1)

>for m = 2 to 30 step 2 @ m, fnb(m) @ next m
 2       .166666666667
 4       -3.33333333333E-2
 6       2.38095238095E-2
 8       -3.33333333333E-2
 10      7.57575757576E-2
 12      -.253113553114
 14      1.16666666667
 16      -7.09215686275
 18      54.9711779449
 20      -529.124242424
 22      6192.12318841
 24      -86580.2531135
 26      1425517.16667
 28      -27298231.0678
 30      601580873.9

>fnb(100)
-2.83822495705E78