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.

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A363293 G.f. A(x) satisfies: A(x) = x * exp( A(x)^2/x - A(-x^2)^2/(2*x^2) + A(x^3)^2/(3*x^3) - A(-x^4)^2/(4*x^4) + ... ).

Original entry on oeis.org

1, 1, 2, 7, 26, 101, 412, 1756, 7692, 34350, 155980, 718312, 3345890, 15735091, 74613107, 356348561, 1712593184, 8276207120, 40192085383, 196045684833, 960042529894, 4718201036195, 23263440797758, 115042992517035, 570463195069614, 2835840294969867, 14129895469191476
Offset: 1

Views

Author

Ilya Gutkovskiy, May 26 2023

Keywords

Crossrefs

Cf. A005750, A005754, A049075, A363294.

Programs

  • Mathematica
    nmax = 27; A[] = 0; Do[A[x] = x Exp[-Sum[A[-(-x)^k]^2/(k (-x)^k), {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

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 11-12 of 12 results.

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

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