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 1-6 of 6 results.

A179469 G.f. satisfies A(x) = exp( Sum_{n>=1} 2^n*A(x^n)*x^n/n ).

Original entry on oeis.org

1, 2, 8, 32, 140, 624, 2928, 14048, 69200, 347040, 1768120, 9122144, 47572128, 250341312, 1327718272, 7089595552, 38082093120, 205638343552, 1115635692576, 6078058719232, 33239328613648, 182402290944576, 1004073853702320
Offset: 0

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Author

Paul D. Hanna, Jul 15 2010

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 32*x^3 + 140*x^4 + 624*x^5 + +...
log(A(x)) = 2*A(x) + 4*A(x^2)*x^2/2 + 8*A(x^3)*x^3/3 + 16*A(x^4)*x^4/4 +...
		

Crossrefs

Programs

  • PARI
    {a(n)=my(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A,x,x^m+x*O(x^n))*2^m*x^m/m)));polcoeff(A,n)}

Formula

From Seiichi Manyama, Jun 02 2023: (Start)
A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-2*x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 2^(k/d) * a(d-1) ) * a(n-k). (End)

A363423 G.f. satisfies A(x) = exp( Sum_{k>=1} A(3*x^k) * x^k/k ).

Original entry on oeis.org

1, 1, 4, 40, 1126, 92440, 22559276, 16468584194, 36033333480881, 236450784546518006, 4654297351684653345788, 274836259327539399144691019, 48686693681325683653963188907344, 25874153864215746591981599665978198380
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2023

Keywords

Crossrefs

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, 3*x^k)*x^k/k)+x*O(x^n))); Vec(A);

Formula

A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(k+1))^(3^k * a(k)).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 3^(d-1) * a(d-1) ) * a(n-k).

A363424 G.f. satisfies A(x) = exp( Sum_{k>=1} A(4*x^k) * x^k/k ).

Original entry on oeis.org

1, 1, 5, 85, 5535, 1422815, 1458410395, 5975113492155, 97902240525033630, 6416219161308006188750, 1681979772433159156954845846, 1763685303864317080584539713676102, 7397434088431352859937186709876628421294
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2023

Keywords

Crossrefs

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, 4*x^k)*x^k/k)+x*O(x^n))); Vec(A);

Formula

A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(k+1))^(4^k * a(k)).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 4^(d-1) * a(d-1) ) * a(n-k).

A363425 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * A(2*x^k) * x^k/k ).

Original entry on oeis.org

1, 1, 2, 10, 89, 1521, 50300, 3271556, 422093896, 108481853032, 55651639993132, 57043042723263188, 116881250986006852062, 478862542730584327952230, 3923320929876295358082556380, 64283613915707884845087288240332
Offset: 0

Views

Author

Seiichi Manyama, Jun 01 2023

Keywords

Crossrefs

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (-1)^(k+1)*subst(A, x, 2*x^k)*x^k/k)+x*O(x^n))); Vec(A);

Formula

A(x) = Sum_{k>=0} a(k) * x^k = Product_{k>=0} (1+x^(k+1))^(2^k * a(k)).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d * 2^(d-1) * a(d-1) ) * a(n-k).

A363481 G.f. satisfies A(x) = exp( 2 * Sum_{k>=1} A(2*x^k) * x^k/k ).

Original entry on oeis.org

1, 2, 11, 108, 1969, 67542, 4473663, 582167944, 150236569819, 77226088637142, 79235069050108841, 162432444097491547308, 665648716390456030366881, 5454326724964994060395500598, 89374602386639273949112262243227
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2023

Keywords

Crossrefs

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(2*sum(k=1, i, subst(A, x, 2*x^k)*x^k/k)+x*O(x^n))); Vec(A);

Formula

A(x) = B(x)^2 where B(x) is the g.f. of A363480.
A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(k+1))^(2^(k+1) * a(k)).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 2^d * a(d-1) ) * a(n-k).

A363480 G.f. satisfies A(x) = exp( Sum_{k>=1} A(2*x^k)^2 * x^k/k ).

Original entry on oeis.org

1, 1, 5, 49, 923, 32603, 2198413, 288677317, 74816592016, 38536646525164, 39578607089767640, 81176446754286348780, 332742981886258629407221, 2726830211640382050679262877, 44684572695377447660556579448947
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2023

Keywords

Crossrefs

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, 2*x^k)^2*x^k/k)+x*O(x^n))); Vec(A);

Formula

G.f.: sqrt(B(x)) where B(x) is the g.f. of A363481.
Showing 1-6 of 6 results.