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 17 results. Next

A384802 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A213109.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 22, 0, 1, 4, 15, 62, 233, 0, 1, 5, 24, 126, 696, 3716, 0, 1, 6, 35, 220, 1497, 11082, 77257, 0, 1, 7, 48, 350, 2768, 24228, 229756, 2026606, 0, 1, 8, 63, 522, 4665, 46004, 504657, 5961846, 63726497, 0, 1, 9, 80, 742, 7368, 80100, 969400, 13042326, 185814320, 2333516392, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2025

Keywords

Examples

			Square array begins:
  1,    1,     1,     1,     1,     1, ...
  0,    1,     2,     3,     4,     5, ...
  0,    3,     8,    15,    24,    35, ...
  0,   22,    62,   126,   220,   350, ...
  0,  233,   696,  1497,  2768,  4665, ...
  0, 3716, 11082, 24228, 46004, 80100, ...

Crossrefs

Columns k=0..1 give A000007, A213109.

Programs

  • PARI
    b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-3*n+3*j+k)^(j-1)*binomial(n, j)*b(n-j, j)));
a(n, k) = b(n, -k);

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-3*n+3*j+k)^(j-1) * binomial(n,j) * b(n-j,j). Then A(n,k) = b(n,-k).

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),", "))

A213112 E.g.f.: A(x) = exp( x/A(-x*A(x)^7)^3 ).

Original entry on oeis.org

1, 1, 7, 118, 2953, 109156, 5220649, 316358470, 23113133089, 1989812691208, 196917302640241, 22027382030604226, 2745173167377165793, 376884883299800082988, 56471832695739964146505, 9164249250078891945300886, 1600258838038369930772797249
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 + 7*x^2/2! + 118*x^3/3! + 2953*x^4/4! + 109156*x^5/5! +...
Related expansions:
A(x)^3 = 1 + 3*x + 27*x^2/2! + 486*x^3/3! + 12825*x^4/4! + 477108*x^5/5! +...
A(x)^7 = 1 + 7*x + 91*x^2/2! + 1918*x^3/3! + 56329*x^4/4! + 2194612*x^5/5! +...
1/A(-x*A(x)^7)^3 = 1 + 3*x + 33*x^2/2! + 603*x^3/3! + 17913*x^4/4! +...
The logarithm of the e.g.f., log(A(x)) = x/A(-x*A(x)^7)^3, begins:
log(A(x)) = x + 6*x^2/2! + 99*x^3/3! + 2412*x^4/4! + 89565*x^5/5! +...

Crossrefs

Cf. A213108, A213109, A213110, A213111, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A^3,x,-x*A^7+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),", "))

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), ", "))
Showing 1-10 of 17 results. Next

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