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.
3.9561842030261697545408021818783008332999988095...
G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 424*x^4 + 7696*x^5 + 173442*x^6 + 4619266*x^7 + 141315896*x^8 + 4874012942*x^9 + 186981188532*x^10 + ... such that A(x) and B = A(x)^(1/2) satisfy A(x) = 1/(2 - B) + ((1+x) - B)/(2 - (1+x)*B)^2 + ((1+x)^2 - B)^2/(2 - (1+x)^2*B)^3 + ((1+x)^3 - B)^3/(2 - (1+x)^3*B)^4 + ((1+x)^4 - B)^4/(2 - (1+x)^4*B)^5 + ((1+x)^5 - B)^5/(2 - (1+x)^5*B)^6 + ... also, A(x) = 1/(2 + B) + ((1+x) + B)/(2 + (1+x)*B)^2 + ((1+x)^2 + B)^2/(2 + (1+x)^2*B)^3 + ((1+x)^3 + B)^3/(2 + (1+x)^3*B)^4 + ((1+x)^4 + B)^4/(2 + (1+x)^4*B)^5 + ((1+x)^5 + B)^5/(2 + (1+x)^5*B)^6 + ... Notice that A(x)^(1/2) is not an integer series, but instead begins A(x)^(1/2) = 1 + 2*(x/4) + 30*(x/4)^2 + 964*(x/4)^3 + 51894*(x/4)^4 + 3807644*(x/4)^5 + 345572460*(x/4)^6 + 36985627016*(x/4)^7 + 4541283789862*(x/4)^8 + 628123762214444*(x/4)^9 + 96578670976842436*(x/4)^10 + ... thus, given the definition, it is remarkable that A(x) should be an integer series.
{a(n) = my(A=[1,1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A)^(1/2) )^m / (2 - (1+x)^m*Ser(A)^(1/2))^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
G.f.: A(x) = 1 + x + 8*x^2 + 90*x^3 + 1336*x^4 + 24406*x^5 + 530234*x^6 + 13410942*x^7 + 388841734*x^8 + 12762735148*x^9 + 469004980720*x^10 + ... RELATED SERIES. A(x)^(1/2) = 1 + 2*(x/4) + 62*(x/4)^2 + 2756*(x/4)^3 + 163574*(x/4)^4 + 11997852*(x/4)^5 + 1047984172*(x/4)^6 + 106571791752*(x/4)^7 + 12417003030694*(x/4)^8 + ... A(x)^(-1/2) = 1 - 2*(x/4) - 58*(x/4)^2 - 2516*(x/4)^3 - 149434*(x/4)^4 - 11055996*(x/4)^5 - 976190180*(x/4)^6 - 100318703592*(x/4)^7 - 11796814729146*(x/4)^8 - ...
{a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); A = Vec( sum(m=0, #A, ( (1+x)^m - Ser(A)^(-1/2) )^m / (2 - (1+x)^m*Ser(A)^(-1/2))^(m+1) ) ) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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