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.
G.f.: A(x) = 1 + 2*x + 3*x^2 + 3*x^3 + 5*x^4 + 39*x^5 + 206*x^6 + 697*x^7 + 1656*x^8 + 3208*x^9 + 8727*x^10 + 41667*x^11 + 192142*x^12 + ...
where
1 = ... + (x^(-3) - 2*x*A(x))^(-3) + (x^(-2) - 2*x*A(x))^(-2) + (x^(-1) - 2*x*A(x))^(-1) + 1 + (x - 2*x*A(x)) + (x^2 - 2*x*A(x))^2 + (x^3 - 2*x*A(x))^3 + ... + (x^n - 2*x*A(x))^n + ...
and
1 = ... + x^(-5)/(x^(-3) + 2*A(x))^3 + x^(-3)/(x^(-2) + 2*A(x))^2 + x^(-1)/(x^(-1) + 2*A(x)) + x + x^3*(x + 2*A(x)) + x^5*(x^2 + 2*A(x))^2 + x^7*(x^3 + 2*A(x))^3 + ... + x^(2*n+1)*(x^n + 2*A(x))^n + ...
also,
1 = ... + x^9*(1 - 2*A(x)/x^2)^3 + x^4*(1 - 2*A(x)/x)^2 + x*(1 - 2*A(x)) + 1 + x/(1 - 2*A(x)*x^2) + x^4/(1 - 2*A(x)*x^3)^2 + x^9/(1 - 2*A(x)*x^4)^3 + ... + x^(n^2)/(1 - 2*A(x)*x^(n+1))^n + ...
further,
1 = ... + x^9*(1 + 2*A(x)/x^2)^2 + x^4*(1 + 2*A(x)/x) + x + 1/(1 + 2*A(x)*x) + x/(1 + 2*A(x)*x^2)^2 + x^4/(1 - 2*A(x)*x^3)^3 + x^9/(1 - 2*A(x)*x^4)^4 + ... + x^(n^2)/(1 + 2*A(x)*x^(n+1))^(n+1) + ...
SPECIFIC VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04720243920412572796492634515550526365563452970121157309...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 2*A)^n = 1.
(V.2) Let A = A(exp(-2*Pi)) = 0.001874436990256710694689538031391789940066981740061145959...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 2*A)^n = 1.
(V.3) Let A = A(-exp(-Pi)) = -0.03971121915244100584186154683625533541823516978831008865...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 2*A)^n = 1.
(V.4) Let A = A(-exp(-2*Pi)) = -0.001860487547859226152163099117755736250804492732905479139...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 2*A)^n = 1.
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 2*x*Ser(A))^m ), #A)/2);A[n+1]}
for(n=0,30,print1(a(n),", "))
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, x^(2*m+1) * (x^m + 2*Ser(A))^m ), #A)/2);A[n+1]}
for(n=0,30,print1(a(n),", "))
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, x^(m^2)/(1 - 2*Ser(A)*x^(m+1))^m ), #A)/2);A[n+1]}
for(n=0,30,print1(a(n),", "))
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, x^(m^2)/(1 + 2*Ser(A)*x^(m+1))^(m+1) ), #A)/2);A[n+1]}
for(n=0,30,print1(a(n),", "))
This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:
A370031: [1, 1, 0, -1, 2, 15, 27, -1, -76, ...];
A355868: [1, 2, 3, 3, 5, 39, 206, 697, 1656, ...];
A370033: [1, 3, 8, 19, 46, 161, 799, 4021, 17932, ...];
A370034: [1, 4, 15, 53, 185, 711, 3270, 17297, 95108, ...];
A370035: [1, 5, 24, 111, 506, 2379, 12083, 67531, 406284, ...];
A370036: [1, 6, 35, 199, 1117, 6335, 37222, 230809, 1515784, ...];
A370037: [1, 7, 48, 323, 2150, 14349, 97431, 681857, 4956116, ...];
A370038: [1, 8, 63, 489, 3761, 28911, 224174, 1768801, 14298852, ...];
A370039: [1, 9, 80, 703, 6130, 53351, 466315, 4118167, 36941188, ...];
A370043: [1, 10, 99, 971, 9461, 91959, 895518, 8775161, 86870264, ...]; ...
...
where the n-th row function R(n,x) satisfies
Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).
{T(n,k) = my(A=[0,1]); for(i=0,k, A = concat(A,0);
A[#A] = polcoeff( sum(m=-sqrtint(#A+1),#A, (x^m - n*Ser(A))^m ) - 1 + (n-2)*sum(m=1,sqrtint(#A+1), x^(m^2) ), #A-1)/n ); A[k+1]}
for(n=1,12, for(k=1,10, print1(T(n,k),", "));print(""))
G.f.: A(x) = 1 + x - x^3 + 2*x^4 + 15*x^5 + 27*x^6 - x^7 - 76*x^8 + 19*x^9 + 719*x^10 + 1687*x^11 + 184*x^12 - 5976*x^13 - 3749*x^14 + 44093*x^15 + ...
where
Sum_{n=-oo..+oo} (x^n - A(x))^n = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04507828029039130528308497098432879536368681539259286273...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - A)^n = (1 + Pi^(1/4)/gamma(3/4))/2 = 1.0432174056066540...
(V.2) Let A = A(exp(-2*Pi)) = 0.001870930061948701432816606547007172908053584772650237678...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - A)^n = (1 + sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 1.00186744274386954...
(V.3) Let A = A(-exp(-Pi)) = -0.04135017416264159536574596265267969182735801577042441264...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - A)^n = (1 + (Pi/2)^(1/4)/gamma(3/4))/2 = 0.9567895690780584107...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001863955401558124515592555303127910405358304631205735085...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - A)^n = (1 + 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 0.99813255728045356...
{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, (x^n - Ser(A))^n ) - sum(n=0,#A, x^(n^2) ), #A-1) ); A[n+1]}
for(n=1,40, print1(a(n),", "))
G.f.: A(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 46*x^4 + 161*x^5 + 799*x^6 + 4021*x^7 + 17932*x^8 + 71311*x^9 + 268639*x^10 + 1045731*x^11 + 4464576*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 3*A(x))^n = 1 - x - x^4 - x^9 - x^16 - x^25 - x^36 - x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04953636800560980886288845724196786482586224709976648461...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 3*A)^n = (3 - Pi^(1/4)/gamma(3/4))/2 = 0.956782594393345992...
(V.2) Let A = A(exp(-2*Pi)) = 0.001877957090194880545086201853719041435355287864597005509...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 3*A)^n = (3 - sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.998132557256130454...
(V.3) Let A = A(-exp(-Pi)) = -0.03819699447470815952471171970837842342724818247967540335...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 3*A)^n = (3 - (Pi/2)^(1/4)/gamma(3/4))/2 = 1.0432104309219415...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001857032573904813918259314464039219802478066024973444789...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 3*A)^n = (3 - 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.001867442719546432...
{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 3*Ser(A))^m ) - 1 + sum(m=1,#A, x^(m^2) ), #A-1)/3 ); A[n+1]}
for(n=1,30, print1(a(n),", "))
G.f.: A(x) = x + 4*x^2 + 15*x^3 + 53*x^4 + 185*x^5 + 711*x^6 + 3270*x^7 + 17297*x^8 + 95108*x^9 + 511258*x^10 + 2653139*x^11 + 13479835*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = 1 - 2*x - 2*x^4 - 2*x^9 - 2*x^16 - 2*x^25 - 2*x^36 - 2*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05211271680112049721451382589099198923178830298930738503...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 4*A)^n = 2 - Pi^(1/4)/gamma(3/4) = 0.913565188786691985...
(V.2) Let A = A(exp(-2*Pi)) = 0.001881490436109324727231096204943046774873234177072692211...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 4*A)^n = 2 - sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.99626511451226...
(V.3) Let A = A(-exp(-Pi)) = -0.03679381086518350821622244996144281973183248006035375080...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 4*A)^n = 2 - (Pi/2)^(1/4)/gamma(3/4) = 1.08642086184388317...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001853590408074327278987912837104527635895010708605840824...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 4*A)^n = 2 - 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.003734885439...
{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 4*Ser(A))^m ) - 1 + 2*sum(m=1,#A, x^(m^2) ), #A-1)/4 ); A[n+1]}
for(n=1,30, print1(a(n),", "))
G.f.: A(x) = x + 5*x^2 + 24*x^3 + 111*x^4 + 506*x^5 + 2379*x^6 + 12083*x^7 + 67531*x^8 + 406284*x^9 + 2531203*x^10 + 15866775*x^11 + 98883303*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 5*A(x))^n = 1 - 3*x - 3*x^4 - 3*x^9 - 3*x^16 - 3*x^25 - 3*x^36 - 3*x^49 - ...
\SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05497127752043377386868704294930896868077597772598908285...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 5*A)^n = (5 - 3*Pi^(1/4)/gamma(3/4) )/2 = 0.870347783180037978...
(V.2) Let A = A(exp(-2*Pi)) = 0.001885037102906729934432374294398706956703235597857256076...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 5*A)^n = (5 - 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.994397671768391...
(V.3) Let A = A(-exp(-Pi)) = -0.03548990756971248576955893224372969073755967165800772531...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 5*A)^n = (5 - 3*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.12963129276582476...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001850160979277236538611428135062916090397865766804127684...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 5*A)^n = (5 - 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.005602328158639...
{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 5*Ser(A))^m ) - 1 + 3*sum(m=1,#A, x^(m^2) ), #A-1)/5 ); A[n+1]}
for(n=1,30, print1(a(n),", "))
G.f.: A(x) = x + 6*x^2 + 35*x^3 + 199*x^4 + 1117*x^5 + 6335*x^6 + 37222*x^7 + 230809*x^8 + 1515784*x^9 + 10423684*x^10 + 73758799*x^11 + 529151547*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 6*A(x))^n = 1 - 4*x - 4*x^4 - 4*x^9 - 4*x^16 - 4*x^25 - 4*x^36 - 4*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05816104948088020874729529058423242784366544822359858088...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 6*A)^n = 3 - 2*Pi^(1/4)/gamma(3/4) = 0.82713037757338397...
(V.2) Let A = A(exp(-2*Pi)) = 0.001888597166059649200752082246148944967408910981759517793...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 6*A)^n = 3 - 2*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.9925302290245218...
(V.3) Let A = A(-exp(-Pi)) = -0.03427512499419794844050440831018295417511284891315471397...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 6*A)^n = 3 - 2*(Pi/2)^(1/4)/gamma(3/4) = 1.172841723687766...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001846744216948148769402996728724142172026226548695349349...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 6*A)^n = 3 - 2*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.007469770878185...
{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 6*Ser(A))^m ) - 1 + 4*sum(m=1,#A, x^(m^2) ), #A-1)/6 ); A[n+1]}
for(n=1,30, print1(a(n),", "))
G.f.: A(x) = x + 7*x^2 + 48*x^3 + 323*x^4 + 2150*x^5 + 14349*x^6 + 97431*x^7 + 681857*x^8 + 4956116*x^9 + 37422943*x^10 + 291763607*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*x - 5*x^4 - 5*x^9 - 5*x^16 - 5*x^25 - 5*x^36 - 5*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06174306640715063509845961016774795661670689719654375131...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 7*A)^n = (7 - 5*Pi^(1/4)/gamma(3/4))/2 = 0.78391297196672996...
(V.2) Let A = A(exp(-2*Pi)) = 0.001892170701611855386420113452656768397809538392272023405...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 7*A)^n = (7 - 5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.99066278628065227...
(V.3) Let A = A(-exp(-Pi)) = -0.03314064170176376172583062314299135400117746157373562359...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 7*A)^n = (7 - 5*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.2160521546097079...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001843340051041967867985717685295689652563446679869985649...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 7*A)^n = (7 - 5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.00933721359773216...
{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 7*Ser(A))^m ) - 1 + 5*sum(m=1,#A, x^(m^2) ), #A-1)/7 ); A[n+1]}
for(n=1,30, print1(a(n),", "))
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