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.

Previous Showing 11-15 of 15 results.

A345233 G.f. A(x) satisfies: A(x) = x + x^2 / exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

Original entry on oeis.org

1, 1, -1, 0, 1, 0, -2, 1, 3, -4, -3, 11, -2, -22, 21, 32, -72, -18, 180, -95, -350, 496, 449, -1542, 125, 3638, -3161, -6393, 12780, 5636, -35993, 14509, 77907, -97880, -116880, 337924, 24514, -869531, 631306, 1692540, -2949009, -1933940, 9035577, -2312868, -21166895
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A007560, A049075, A345232.

Programs

  • Mathematica
    nmax = 45; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^k A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 45}]

Formula

G.f.: x + x^2 / Product_{n>=1} (1 + x^n)^a(n).
a(n+2) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * a(d) ) * a(n-k+2).

A346031 G.f. A(x) satisfies: A(x) = x + x^3 * exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 1, 1, 4, 4, 3, 9, 14, 12, 22, 43, 49, 66, 130, 186, 234, 406, 663, 884, 1362, 2303, 3347, 4884, 8049, 12478, 18240, 28853, 46075, 69163, 106470, 170305, 262853, 401773, 635780, 998609, 1536093, 2405345, 3801601, 5910267, 9212253, 14548179, 22818301
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 01 2021

Keywords

Crossrefs

Cf. A007560, A316075, A346032.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, [1, 0, 1][n], add(a(n-k)*add(
     (-1)^(k/d+1)*d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..48);  # Alois P. Heinz, Jul 01 2021
  • Mathematica
    nmax = 48; A[] = 0; Do[A[x] = x + x^3 Exp[Sum[(-1)^(k + 1) A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[2] = 0; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 48}]

Formula

G.f.: x + x^3 * Product_{n>=1} (1 + x^n)^a(n).
a(1) = 1, a(2) = 0, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k).

A346032 G.f. A(x) satisfies: A(x) = x^2 + x^3 * exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 0, 2, 1, 1, 4, 2, 4, 9, 4, 14, 20, 15, 43, 48, 55, 127, 127, 199, 363, 379, 684, 1048, 1229, 2263, 3100, 4163, 7288, 9558, 14231, 23222, 30673, 48404, 74113, 101631, 163048, 239282, 343196, 545318, 785139, 1169148, 1818866, 2619072, 3991888, 6079434
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 01 2021

Keywords

Crossrefs

Cf. A007560, A316075, A346031.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, signum(n-1), add(a(n-k)*add(
     (-1)^(k/d+1)*d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 01 2021
  • Mathematica
    nmax = 50; A[] = 0; Do[A[x] = x^2 + x^3 Exp[Sum[(-1)^(k + 1) A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 0; a[2] = 1; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 50}]

Formula

G.f.: x^2 + x^3 * Product_{n>=1} (1 + x^n)^a(n).
a(1) = 0, a(2) = 1, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k).

A363386 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / k ).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 4, 8, 10, 22, 50, 77, 160, 343, 622, 1250, 2648, 5127, 10364, 21685, 43594, 88907, 185458, 380113, 782902, 1633841, 3387444, 7033401, 14716304, 30734066, 64228198, 134824862, 283040684, 594516622, 1252151812, 2639220817, 5566237724, 11760037378
Offset: 1

Views

Author

Ilya Gutkovskiy, May 30 2023

Keywords

Crossrefs

Cf. A005754, A007560, A363385.

Programs

  • Mathematica
    nmax = 38; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^2/k, {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d g[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 38}]
  • PARI
    seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p,x)-1); p = x + x^2*exp(-sum(k=1, m\2, (-1)^k*subst(p + O(x^(m\k+1)), x, x^k)^2/k))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

A363087 G.f. A(x) satisfies: A(x) = x - x^2 * exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...).

Original entry on oeis.org

1, -1, -1, 1, 2, -1, -5, -1, 11, 10, -21, -39, 30, 126, 4, -354, -261, 834, 1347, -1483, -5033, 823, 15663, 8765, -41112, -56364, 84888, 234546, -91319, -791833, -293380, 2251507, 2561264, -5177875, -11835968, 7620048, 42944358, 7464956, -130615874, -119900209
Offset: 1

Views

Author

Ilya Gutkovskiy, May 18 2023

Keywords

Crossrefs

Cf. A007560, A049075, A345234, A363062.

Programs

  • Mathematica
    nmax = 40; A[] = 0; Do[A[x] = x - x^2 Exp[Sum[(-1)^(k + 1) A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[2] = -1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 40}]

Formula

G.f.: x - x^2 * Product_{n>=1} (1 + x^n)^a(n).
a(1) = 1, a(2) = -1; a(n) = (1/(n - 2)) * Sum_{k=1..n-2} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k).
Previous Showing 11-15 of 15 results.

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

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