chiseq[x_]:=Expand[FunctionExpand[-4Catalan/Pi +(2^(2(x)-3)*x^2)/((x-1)!)*Sum[(-1/2)^k*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][k,x-1,x/2]*(Derivative[1,0][Zeta][k-x+2,x/4] - Derivative[1,0][Zeta][k-x+2,(x+2)/4]),{k,0,x-1}]-(2^(2(x)+1))/((x-1)!)*Sum[(-1/2)^k*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][k,x+1,(x+2)/2]*(Derivative[1,0][Zeta][k-x,(x+2)/4] - Derivative[1,0][Zeta][k-x,(x+4)/4]),{k,0,x+1}]]]

chiseq[2]

N[-((4 Catalan)/π)+((14 Zeta[3])/(π^2))]

(∫_0^∞)((Sech[x]-((Sech[x])^2))/(x^2))\[DifferentialD]x

NIntegrate[((Sech[x]-((Sech[x])^2))/(x^2)),{x,0,Infinity}]

chioddseq[x_]:=Expand[FunctionExpand[(-4Catalan)/Pi +((2^(4(x)-1)*(2x+1)^2)/((2x)!)*Sum[1/(4^k)*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][2k,2x,(2x+1)/2]*(Derivative[1,0][Zeta][2k-2x+1,(2x+1)/4] - Derivative[1,0][Zeta][2k-2x+1,(2x+3)/4]),{k,0,x}])-2^(4(x)+3)/((2x)!)*Sum[1/(4^k)*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][2k,2x+2,(2x+3)/2]*(Derivative[1,0][Zeta][2k-2x-1,(2x+3)/4] - Derivative[1,0][Zeta][2k-2x-1,(2x+5)/4]),{k,0,x+1}]]]

chioddseq[3]

deltaseq[x_]:=Expand[FunctionExpand[(2^(2(x)-1)*(x+1)^2)/((x)!)*Sum[(-1/2)^k*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][k,x,(x+1)/2]*(Derivative[1,0][Zeta][k-x+1,(x+1)/4] - Derivative[1,0][Zeta][k-x+1,(x+3)/4]),{k,0,x}]-(2^(2(x)+3))/((x)!)*Sum[(-1/2)^k*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][k,x+2,(x+3)/2]*(Derivative[1,0][Zeta][k-x-1,(x+3)/4] - Derivative[1,0][Zeta][k-x-1,(x+5)/4]),{k,0,x+2}]-(2^(2(x)-3)*x^2)/((x-1)!)*Sum[(-1/2)^k*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][k,x-1,x/2]*(Derivative[1,0][Zeta][k-x+2,x/4] - Derivative[1,0][Zeta][k-x+2,(x+2)/4]),{k,0,x-1}]+(2^(2(x)+1))/((x-1)!)*Sum[(-1/2)^k*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][k,x+1,(x+2)/2]*(Derivative[1,0][Zeta][k-x,(x+2)/4] - Derivative[1,0][Zeta][k-x,(x+4)/4]),{k,0,x+1}] ]]

deltaseq[3]

lambdaeven[x_]:=Expand[FunctionExpand[-2^(4(x))/(x*(2(x)-1)!) *Sum[(1/4^k*(x^2*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][2k-2,2(x)-1,x]+Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][2k,2(x)+1,x+1]))*(Derivative[1,0][Zeta][-2(x)+2k,(x+1)/2]-Derivative[1,0][Zeta][-2(x)+2k,(x)/2]),{k,1,x}]+(2^(2(x))*x^(2x-1)*Log[x/2])/(((2(x)-1)!))*Sum[(1/x^(2k)*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][2k,2(x)+1,x+1]),{k,1,x}]-2^(4(x))/(x*(2(x)-1)!) *(Derivative[1,0][Zeta][-2(x),(x+1)/2]-Derivative[1,0][Zeta][-2(x),(x+2)/2])]]

lambdaeven[1]

lambdaodd[x_]:=Expand[FunctionExpand[((2^(4(x)-1)*(2x+1))/((2x)!)*Sum[1/(4^k)*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][2k,2x,(2x+1)/2]*(Derivative[1,0][Zeta][2k-2x+1,(2x+1)/4] - Derivative[1,0][Zeta][2k-2x+1,(2x+3)/4]),{k,0,x}])-2^(4(x)+3)/((2x+1)!)*Sum[1/(4^k)*Function[Sum[(-1)^(i-#1) #3^i Binomial[i+#2-#1, #2-#1] Abs[StirlingS1[#2+1,i+#2-#1+1]],{i,0,#1}]][2k,2x+2,(2x+3)/2]*(Derivative[1,0][Zeta][2k-2x-1,(2x+3)/4] - Derivative[1,0][Zeta][2k-2x-1,(2x+5)/4]),{k,0,x+1}]]]

lambdaodd[1]