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 1-10 of 18 results. Next

A213108 E.g.f.: A(x) = exp( x/A(-x*A(x)) ).

Original entry on oeis.org

1, 1, 3, 10, 41, 76, -2183, -54998, -1045567, -15948296, -157645999, 2035442014, 217585291057, 10000385378452, 373813151971001, 11759936127330346, 269243105500780673, -519586631788126352, -649842878319124373855, -59793494397006229506890
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare the e.g.f. to:
(1) W(x) = exp(x/W(-x*W(x)^2)^1) when W(x) = Sum_{n>=0} (1*n+1)^(n-1)*x^n/n!.
(2) W(x) = exp(x/W(-x*W(x)^4)^2) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.
(3) W(x) = exp(x/W(-x*W(x)^6)^3) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 41*x^4/4! + 76*x^5/5! - 2183*x^6/6! +...
Related expansions:
1/A(-x*A(x)) = 1 + x + x^2/2! + x^3/3! - 23*x^4/4! - 419*x^5/5! - 5159*x^6/6! +...
The logarithm of the e.g.f., log(A(x)) = x/A(-x*A(x)), begins:
log(A(x)) = x + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! - 115*x^5/5! - 2514*x^6/6! - 36113*x^7/7! +...

Crossrefs

Cf. A213109, A213110, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A,x,-x*A+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A213113 E.g.f.: A(x) = exp( x/A(-x*A(x)^9)^3 ).

Original entry on oeis.org

1, 1, 7, 154, 4681, 228076, 14299129, 1138327282, 108153498625, 11945906543512, 1500579818594641, 210620216812835446, 32619162944121580369, 5512919937646519781956, 1007971183370936380058233, 197907153405452704613136466, 41467801090663272520003650049
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare the e.g.f. to:
(1) W(x) = exp(x/W(-x*W(x)^2)^1) when W(x) = Sum_{n>=0} (1*n+1)^(n-1)*x^n/n!.
(2) W(x) = exp(x/W(-x*W(x)^4)^2) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.
(3) W(x) = exp(x/W(-x*W(x)^6)^3) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.
(4) W(x) = exp(x/W(-x*W(x)^8)^4) when W(x) = Sum_{n>=0} (4*n+1)^(n-1)*x^n/n!.

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 154*x^3/3! + 4681*x^4/4! + 228076*x^5/5! +...
Related expansions:
A(x)^3 = 1 + 3*x + 27*x^2/2! + 594*x^3/3! + 18873*x^4/4! + 902988*x^5/5! +...
A(x)^9 = 1 + 9*x + 135*x^2/2! + 3402*x^3/3! + 121257*x^4/4! + 5887404*x^5/5! +...
1/A(-x*A(x)^9)^3 = 1 + 3*x + 45*x^2/2! + 999*x^3/3! + 39609*x^4/4! +...
The logarithm of the e.g.f., log(A(x)) = x/A(-x*A(x)^9)^3, begins:
log(A(x)) = x + 6*x^2/2! + 135*x^3/3! + 3996*x^4/4! + 198045*x^5/5! +...

Crossrefs

Cf. A213108, A213109, A213110, A213111, A213112.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A^3,x,-x*A^9+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A213109 E.g.f.: A(x) = exp( x/A(-x*A(x)^3) ).

Original entry on oeis.org

1, 1, 3, 22, 233, 3716, 77257, 2026606, 63726497, 2333516392, 97335801521, 4543398147674, 234240366949921, 13191513757571644, 804299893048589225, 52696560194440470046, 3686739789058021079873, 273950438842854064788560, 21522076959435116533294177
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare the e.g.f. to:
(1) W(x) = exp(x/W(-x*W(x)^2)^1) when W(x) = Sum_{n>=0} (1*n+1)^(n-1)*x^n/n!.
(2) W(x) = exp(x/W(-x*W(x)^4)^2) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.
(3) W(x) = exp(x/W(-x*W(x)^6)^3) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 233*x^4/4! + 3716*x^5/5! +...
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2/2! + 126*x^3/3! + 1497*x^4/4! + 24228*x^5/5! +...
1/A(-x*A(x)^3) = 1 + x + 5*x^2/2! + 37*x^3/3! + 489*x^4/4! + 8541*x^5/5! +...
The logarithm of the e.g.f., log(A(x)) = x/A(-x*A(x)^3), begins:
log(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 148*x^4/4! + 2445*x^5/5! + 51246*x^6/6! +...

Crossrefs

Cf. A213108, A213110, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A,x,-x*A^3+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A213110 E.g.f.: A(x) = exp( x/A(-x*A(x)^5)^2 ).

Original entry on oeis.org

1, 1, 5, 61, 1089, 29081, 1006753, 44229669, 2338846849, 145278355825, 10340497436481, 829144792315709, 73858518558797569, 7228342584930637353, 770235745321038739681, 88690109534418912004501, 10965585032265975064491777, 1447844650991790389918127329
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare the e.g.f. to:
(1) W(x) = exp(x/W(-x*W(x)^2)^1) when W(x) = Sum_{n>=0} (1*n+1)^(n-1)*x^n/n!.
(2) W(x) = exp(x/W(-x*W(x)^4)^2) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.
(3) W(x) = exp(x/W(-x*W(x)^6)^3) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 61*x^3/3! + 1089*x^4/4! + 29081*x^5/5! +...
Related expansions:
A(x)^2 = 1 + 2*x + 12*x^2/2! + 152*x^3/3! + 2816*x^4/4! + 75152*x^5/5! +...
A(x)^5 = 1 + 5*x + 45*x^2/2! + 665*x^3/3! + 13745*x^4/4! + 380525*x^5/5! +...
1/A(-x*A(x)^5)^2 = 1 + 2*x + 16*x^2/2! + 206*x^3/3! + 4456*x^4/4! +...
The logarithm of the e.g.f., log(A(x)) = x/A(-x*A(x)^5)^2, begins:
log(A(x)) = x + 4*x^2/2! + 48*x^3/3! + 824*x^4/4! + 22280*x^5/5! + 774012*x^6/6! +...

Crossrefs

Cf. A213108, A213109, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A^2,x,-x*A^5+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A213111 E.g.f.: A(x) = exp( x/A(-x*A(x)^6)^2 ).

Original entry on oeis.org

1, 1, 5, 73, 1497, 48321, 2016733, 106687113, 6745180529, 495988880833, 41495596689141, 3880618840698249, 400537444634948041, 45126092520882513921, 5501154522933362385485, 720279890636684703825481, 100658531630809161730405857, 14934726665907895887483076737
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare the e.g.f. to:
(1) W(x) = exp(x/W(-x*W(x)^2)^1) when W(x) = Sum_{n>=0} (1*n+1)^(n-1)*x^n/n!.
(2) W(x) = exp(x/W(-x*W(x)^4)^2) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.
(3) W(x) = exp(x/W(-x*W(x)^6)^3) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 1497*x^4/4! + 48321*x^5/5! +...
Related expansions:
A(x)^2 = 1 + 2*x + 12*x^2/2! + 176*x^3/3! + 3728*x^4/4! + 118912*x^5/5! +...
A(x)^6 = 1 + 6*x + 60*x^2/2! + 1008*x^3/3! + 23952*x^4/4! + 775296*x^5/5! +...
1/A(-x*A(x)^6)^2 = 1 + 2*x + 20*x^2/2! + 296*x^3/3! + 7824*x^4/4! +...
The logarithm of the e.g.f., log(A(x)) = x/A(-x*A(x)^6)^2, begins:
log(A(x)) = x + 4*x^2/2! + 60*x^3/3! + 1184*x^4/4! + 39120*x^5/5! + 1639872*x^6/6! +...

Crossrefs

Cf. A213108, A213109, A213110, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A^2,x,-x*A^6+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A213225 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^4)).

Original entry on oeis.org

1, 1, 2, 6, 20, 76, 313, 1375, 6337, 30243, 148129, 739172, 3737993, 19077868, 97955307, 504707999, 2604312205, 13436676965, 69229324721, 355854322633, 1823672937884, 9314227843463, 47406130512872, 240498260267049, 1216833204738419, 6146116088495029, 31030233400282749
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2012

Keywords

Comments

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 76*x^5 + 313*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 201*x^4 + 816*x^5 + 3468*x^6 +...
1/A(-x*A(x)^4) = 1 + x + 3*x^2 + 9*x^3 + 35*x^4 + 146*x^5 + 656*x^6 +...

Crossrefs

Cf. A213226, A213227, A213228, A213229, A213230, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

Programs

  • Mathematica
    terms = 26; A[] = 1; Do[A[x] = 1/(1-x/A[-x*A[x]^4]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Aug 23 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A, x, -x*subst(A^4, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213226 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^5)).

Original entry on oeis.org

1, 1, 2, 7, 27, 122, 607, 3208, 17688, 99803, 571238, 3292738, 19001315, 109303307, 624615928, 3537913240, 19843769848, 110273489737, 608712132055, 3355449334452, 18624818099047, 105191779542849, 610586100129734, 3662333209225714, 22652502251884322
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2012

Keywords

Comments

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 122*x^5 + 607*x^6 +...
Related expansions:
A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 380*x^4 + 1801*x^5 + 9045*x^6 +...
1/A(-x*A(x)^5) = 1 + x + 4*x^2 + 14*x^3 + 66*x^4 + 336*x^5 + 1805*x^6 +...

Crossrefs

Cf. A213225, A213227, A213228, A213229, A213230, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A, x, -x*subst(A^5, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213228 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^6)^2).

Original entry on oeis.org

1, 1, 3, 14, 73, 440, 2862, 19991, 146939, 1125413, 8896018, 72067978, 595097838, 4987609871, 42290465703, 361845473658, 3117830204185, 27009650432888, 234932107635587, 2049479335366836, 17915253987741538, 156799716352350344, 1373180896765862962
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2012

Keywords

Comments

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 73*x^4 + 440*x^5 + 2862*x^6 +...
Related expansions:
A(x)^6 = 1 + 6*x + 33*x^2 + 194*x^3 + 1188*x^4 + 7656*x^5 + 51583*x^6 +...
1/A(-x*A(x)^6)^2 = 1 + 2*x + 9*x^2 + 44*x^3 + 268*x^4 + 1750*x^5 + 12422*x^6 +...

Crossrefs

Cf. A213225, A213226, A213227, A213229, A213230, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^2, x, -x*subst(A^6, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213229 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^7)^2).

Original entry on oeis.org

1, 1, 3, 16, 93, 649, 4924, 40221, 344817, 3058115, 27798895, 257009431, 2404734586, 22679499148, 214947515333, 2042353663088, 19417906390395, 184458621283607, 1748712359825873, 16530801697256737, 155736745914813741, 1461877902947680987, 13674142992787617967
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2012

Keywords

Comments

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 93*x^4 + 649*x^5 + 4924*x^6 +...
Related expansions:
A(x)^7 = 1 + 7*x + 42*x^2 + 273*x^3 + 1862*x^4 + 13531*x^5 + 104062*x^6 +...
1/A(-x*A(x)^7)^2 = 1 + 2*x + 11*x^2 + 60*x^3 + 431*x^4 + 3302*x^5 + 27421*x^6 +..

Crossrefs

Cf. A213225, A213226, A213227, A213228, A213230, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^2, x, -x*subst(A^7, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213230 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^8)^2).

Original entry on oeis.org

1, 1, 3, 18, 115, 902, 7722, 70784, 678251, 6670586, 66851992, 677328214, 6903177354, 70490174298, 718856047396, 7304677030708, 73837797474235, 741722190452840, 7402780597473820, 73459355234486763, 726095774886910232, 7170907377415662763, 71063833561266044578
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2012

Keywords

Comments

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 115*x^4 + 902*x^5 + 7722*x^6 +...
Related expansions:
A(x)^8 = 1 + 8*x + 52*x^2 + 368*x^3 + 2754*x^4 + 22112*x^5 + 189344*x^6 +...
1/A(-x*A(x)^8)^2 = 1 + 2*x + 13*x^2 + 78*x^3 + 634*x^4 + 5488*x^5 + 50969*x^6 +...

Crossrefs

Cf. A213225, A213226, A213227, A213228, A213229, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^2, x, -x*subst(A^8, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-10 of 18 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.