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 1-4 of 4 results.

A363541 G.f. satisfies A(x) = exp( Sum_{k>=1} (3^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 4, 17, 73, 324, 1469, 6838, 32490, 157398, 775010, 3870690, 19567202, 99957231, 515250057, 2676884745, 14002926871, 73693381322, 389904743248, 2072794614996, 11066421965311, 59310040841395, 318978744562253, 1720962766007827
Offset: 0

Views

Author

Seiichi Manyama, Jun 09 2023

Keywords

Crossrefs

Programs

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

Formula

A(x) = B(x)/(1 - 3*x) where B(x) is the g.f. of A363546.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-3*x) * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 3^k + Sum_{d|k} d * a(d-1) ) * a(n-k).

A363509 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (3 + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 4, 10, 30, 101, 361, 1354, 5238, 20740, 83683, 342719, 1421019, 5953306, 25162342, 107163924, 459438524, 1981247950, 8588054014, 37398421941, 163534601567, 717776072291, 3161117717887, 13964782042188, 61866495037806, 274792382789958
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2023

Keywords

Crossrefs

Programs

  • Mathematica
    terms = 25; A[] = 0; Do[A[x] = Exp[Sum[(-1)^(k+1)*(3+A[x^k])*x^k/k,{k,terms}]]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, May 10 2025 *)
  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (-1)^(k+1)*(3+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);

Formula

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

A363508 G.f. satisfies A(x) = exp( Sum_{k>=1} (4 + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 5, 20, 80, 340, 1516, 7046, 33736, 165436, 826566, 4193348, 21542664, 111848161, 585949358, 3093526496, 16442687695, 87914559018, 472522551440, 2551591234444, 13836226412386, 75311992329508, 411336641019998, 2253641429297336
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2023

Keywords

Crossrefs

Programs

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

Formula

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

A363548 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - x^k)^3) ).

Original entry on oeis.org

1, 1, 5, 19, 79, 326, 1414, 6198, 27794, 126233, 580885, 2700135, 12665756, 59869222, 284919675, 1364009722, 6564545500, 31742029545, 154134718727, 751316355122, 3674923035139, 18031965040197, 88734141475113, 437813286219942, 2165445447313147
Offset: 0

Views

Author

Seiichi Manyama, Jun 09 2023

Keywords

Crossrefs

Programs

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

Formula

A(x) = (1 - x)^3 * B(x) where B(x) is the g.f. of A363507.
a(n) = Sum_{k=0..3} (-1)^k * binomial(3,k) * A363507(n-k).
Showing 1-4 of 4 results.