cp's OEIS Frontend

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.

Showing 1-8 of 8 results.

A379201 Column 1 of triangle A379200; a(n) = A379200(n,1) for n >= 2.

Original entry on oeis.org

1, 4, 13, 40, 130, 472, 1863, 7536, 30144, 118420, 460746, 1795688, 7059729, 28023636, 111978480, 448799712, 1800225540, 7223348112, 29006342666, 116643481440, 469861180529, 1895748851480, 7658862165738, 30973386496992, 125363177936450, 507777786429164, 2058225234116544
Offset: 2

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

G.f. G(x,y) of triangle A379200 satisfies 1/x = Sum_{n=-oo..+oo} G(x,y)^n * (G(x,y)^n + y)^(n+1). This sequence gives the coefficients of x^n*y in G(x,y) for n >= 2.

Links

Crossrefs

Cf. A379200.

A379206 Central terms of triangle A379200; a(n) = A379200(2*n-1,n-1) for n >= 1.

Original entry on oeis.org

1, 4, 52, 1004, 24540, 693528, 21365548, 694033712, 23369007288, 808414759404, 28582681491280, 1029170281603296, 37633688114705676, 1394397634883242504, 52252719103306019248, 1977346544751795430944, 75468918458797503960180, 2902157847756806886385760, 112350085062188369503742656
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

G.f. G(x,y) of triangle A379200 satisfies 1/x = Sum_{n=-oo..+oo} G(x,y)^n * (G(x,y)^n + y)^(n+1). This sequence gives the coefficients of x^(2*n-1)*y^(n-1) in G(x,y) for n >= 1.

Links

Crossrefs

Cf. A379200.

A378264 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).

Original entry on oeis.org

1, 3, 10, 38, 164, 783, 4005, 21400, 117602, 659019, 3748736, 21588796, 125646501, 737977155, 4369147468, 26048215099, 156249597852, 942344615209, 5710710976884, 34756875588376, 212361179832431, 1302068876523950, 8009024360554817, 49407447276951470, 305609996146288873, 1895015255546957578
Offset: 1

Views

Author

Paul D. Hanna, Dec 08 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Examples

			G.f.: A(x) = x + 3*x^2 + 10*x^3 + 38*x^4 + 164*x^5 + 783*x^6 + 4005*x^7 + 21400*x^8 + 117602*x^9 + 659019*x^10 + 3748736*x^11 + 21588796*x^12 + ...
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.14832728317680424382350400745104642263167027946862...
A(t) = 1/4 at t = 0.13433913917600443178696714330960568436967435856815...
A(t) = 1/5 at t = 0.12029812285398972879219940261295281978412524937754...
A(3/20) = 0.3521325903099608361455770617898033111722103407971...
A(1/7) = 0.29252723487814042698570516039406838227427731852655...
A(1/8) = 0.21500724214149512130643660913381998900575603076452...
A(1/9) = 0.17407688053908806913569913139334508111874650183559...
A(1/10) = 0.14711097488062849474543678333471254427936118296317...

Links

Crossrefs

Cf. A379200, A379199, A166952, A379202, A379203, A379204, A379205.

Programs

  • PARI
    {a(n) = my(V=[0,1],A); for(i=1,n, V=concat(V,0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A,#A, A^m*(1 + A^m)^(m+1) ), #V-3); ); polcoef(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).
(2) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + A(x)^(n+1))^n.
From Paul D. Hanna, Dec 20 2024: (Start)
(3) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 1)^n.
(4) A(x) = x * Sum_{n=-oo..+oo, n <> -1} A(x)^(n^2) / (1 - A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 1)^(n+1) ).
(End)

A379199 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n - 1)^(n+1).

Original entry on oeis.org

1, 1, 2, 2, 4, 9, 45, 164, 546, 1493, 3944, 10588, 32997, 112945, 396404, 1330461, 4265180, 13292275, 41778612, 135378928, 452828655, 1534394542, 5175561385, 17246318586, 56998526633, 188492707958, 628391304843, 2115131897264, 7162685531894, 24280930956521, 82152859633099
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 45*x^7 + 164*x^8 + 546*x^9 + 1493*x^10 + 3944*x^11 + 10588*x^12 + ...
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.28045847462385815185359630099816126187110099265378...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (2^(n-1) - 1)^n / 2^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (2^(n-1) + 1)^(n-1) / 2^(n^2-1).
A(t) = 1/3 at t = 0.23482705460970305955617199360925350115096428519729...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (3^(n-1) - 1)^n / 3^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (3^(n-1) + 1)^(n-1) / 3^(n^2-1).
A(t) = 1/4 at t = 0.19291797602834900465339136778069433360676297133766...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (4^(n-1) - 1)^n / 4^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (4^(n-1) + 1)^(n-1) / 4^(n^2-1).
A(1/4) = 0.37094847513809700088242935848658292140487254454012...
  where 4 = Sum_{n=-oo..+oo} A(1/4)^n * (A(1/4)^n - 1)^(n+1),
  also, 4 = Sum_{n=-oo..+oo} A(1/4)^(2*n) * (A(1/4)^n + 1)^n.
A(1/5) = 0.26269124124750053890427847522296583687631694884657...
A(1/6) = 0.20631303406093749454201994379654348907240460444958...
A(1/7) = 0.17034902087146833005156413354158308643804109633470...
A(1/8) = 0.14521334319041207588863463072178319621820854479438...

Links

Crossrefs

Cf. A379200, A166952, A378264, A379202, A379203, A379204, A379205.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m - 1)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n - 1)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n + 1)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + A(x)^(n+1))^(n+1).
(4) A(x) = x * Sum_{n=-oo..+oo, n <> -1} A(x)^(n^2) / (1 - A(x)^(n+1))^n.
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n - 1)^(n+1) ).

A379202 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).

Original entry on oeis.org

1, 4, 20, 122, 850, 6432, 51324, 424694, 3608592, 31291658, 275774228, 2462835772, 22239367632, 202713590686, 1862689951724, 17235880764264, 160466865121154, 1502055108051124, 14127846520455180, 133455751612975948, 1265563747442829216, 12043611154775588194, 114978748131733714360
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
Conjecture: a(n) is even for n > 1.
It appears that a(n) == 2 (mod 4) at n = A028309(k) for k >= 4.

Examples

			G.f.: A(x) = x + 4*x^2 + 20*x^3 + 122*x^4 + 850*x^5 + 6432*x^6 + 51324*x^7 + 424694*x^8 + 3608592*x^9 + 31291658*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.090270773138940793847220645261976952310511883470512...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*6^(n-1))^n / 6^(n^2-1).
A(t) = 1/7 at t = 0.084362907984862824662513569761745773472320783010611...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*7^(n-1))^n / 7^(n^2-1).
A(t) = 1/8 at t = 0.078703999402417120618295617221021413542415048822164...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*8^(n-1))^n / 8^(n^2-1).
A(1/11) = 0.16976727159020613475135380983780463368461713164010...
A(1/12) = 0.13933682309394427848416123650354034389806333559384...
A(1/15) = 0.09515898887066227963795425335824195002284059150209...
A(1/20) = 0.06369786461564277053938913595571090186089127528505...

Links

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379203, A379204, A379205.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 2)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 2)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 2*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 2*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 2)^(n+1) ).

A379203 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 3)^(n+1).

Original entry on oeis.org

1, 5, 34, 290, 2820, 29629, 327301, 3744868, 43981858, 527126689, 6420981368, 79260797860, 989306411413, 12464737320229, 158320378037652, 2025016002188169, 26060398562711196, 337197048402240367, 4384067953773647268, 57245716462267462224, 750403639664344374239, 9871281245683966836462
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Examples

			G.f.: A(x) = x + 5*x^2 + 34*x^3 + 290*x^4 + 2820*x^5 + 29629*x^6 + 327301*x^7 + 3744868*x^8 + 43981858*x^9 + 527126689*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/7 at t = 0.069769772400266469707360138034033927488705716660080...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*7^(n-1))^n / 7^(n^2-1).
A(t) = 1/8 at t = 0.067295105779482404156544832668824160420208234924667...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*8^(n-1))^n / 8^(n^2-1).
A(t) = 1/9 at t = 0.064327556053208007320009998534415581932268509899202...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*9^(n-1))^n / 9^(n^2-1).
A(t) = 1/10 at t = 0.06126924119589872239866986020862532219839002819792...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*10^(n-1))^n / 10^(n^2-1).
A(1/15) = 0.12166176397390884847529063617720403039492284665035...
A(1/16) = 0.10420546336336096378642246758350885785023968035181...
A(1/20) = 0.07053009254165709187694647754531300907207762301254...

Links

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379204, A379205.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 3)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 3)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 3)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 3*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 3*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 3)^(n+1) ).

A379204 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).

Original entry on oeis.org

1, 6, 52, 572, 7154, 96444, 1365480, 20015404, 301104656, 4622137698, 72110068424, 1140008607808, 18223311950352, 294049155429240, 4783093039542544, 78348659072215696, 1291254702576739650, 21396346604365855060, 356250789435149406252, 5957201829333106382128, 100003077199160840926640
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
Conjecture: a(n) is even for n > 1.
Conjecture: a(n) == 2 (mod 4) iff n = (k-1)^2 + 1 for some k > 1.

Examples

			G.f.: A(x) = x + 6*x^2 + 52*x^3 + 572*x^4 + 7154*x^5 + 96444*x^6 + 1365480*x^7 + 20015404*x^8 + 301104656*x^9 + 4622137698*x^10 + ...

Links

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379203, A379205.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 4)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 4)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 4*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 4*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 4)^(n+1) ).

A379205 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 5)^(n+1).

Original entry on oeis.org

1, 7, 74, 998, 15268, 251427, 4345869, 77751128, 1427455842, 26740178711, 509068777424, 9820550568868, 191554931918517, 3771529984556599, 74857068226445132, 1496158969938529383, 30086862802675119068, 608303992207446069349, 12358069554479794052292, 252144178158939689795128
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Examples

			G.f.: A(x) = x + 7*x^2 + 74*x^3 + 998*x^4 + 15268*x^5 + 251427*x^6 + 4345869*x^7 + 77751128*x^8 + 1427455842*x^9 + 26740178711*x^10 + ...

Links

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379203, A379204.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 5)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 5)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 5)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 5*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 5*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 5)^(n+1) ).
Showing 1-8 of 8 results.

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