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 1-3 of 3 results.

A306768 G.f. A(x) satisfies: A(x) = x*exp(-A(-x) + A(-x^2)/2 - A(-x^3)/3 + A(-x^4)/4 - A(-x^5)/5 + ...).

Original entry on oeis.org

0, 1, 1, -1, -2, 2, 6, -5, -18, 15, 59, -54, -215, 199, 813, -744, -3135, 2890, 12394, -11538, -50017, 46806, 204893, -192451, -849681, 800974, 3560927, -3367656, -15058478, 14279426, 64171736, -60992032, -275304665, 262199050, 1188070488, -1133572891, -5153913606
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 14 2019

Keywords

Examples

			G.f.: A(x) = x + x^2 - x^3 - 2*x^4 + 2*x^5 + 6*x^6 - 5*x^7 - 18*x^8 + 15*x^9 + 59*x^10 - 54*x^11 - 215*x^12 + ...

Crossrefs

Cf. A000081, A004111, A045648, A049075, A073075, A115593, A307365, A307366, A307538.

Programs

  • Mathematica
    terms = 36; A[] = 0; Do[A[x] = x Exp[Sum[(-1)^k A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 + x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 36}]

Formula

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

A307365 G.f. A(x) satisfies: A(x) = x*exp(A(-x) + A(-x^2)/2 + A(-x^3)/3 + A(-x^4)/4 + ...).

Original entry on oeis.org

0, 1, -1, -1, 2, 1, -4, -3, 11, 10, -36, -32, 122, 105, -420, -368, 1497, 1336, -5491, -4919, 20477, 18393, -77397, -69883, 296306, 268711, -1146538, -1042924, 4475265, 4081598, -17600475, -16091719, 69681964, 63845971, -277494594, -254730047, 1110782803, 1021361912
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2019

Keywords

Examples

			G.f.: A(x) = x - x^2 - x^3 + 2*x^4 + x^5 - 4*x^6 - 3*x^7 + 11*x^8 + 10*x^9 - 36*x^10 - 32*x^11 + ...

Crossrefs

Cf. A000081, A004111, A045648, A049075, A307366.

Programs

  • Mathematica
    terms = 37; A[] = 0; Do[A[x] = x Exp[Sum[A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 37}]

Formula

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

A307538 G.f. A(x) satisfies: A(x) = x*exp(2*A(-x) + 2*A(-x^3)/3 + 2*A(-x^5)/5 + 2*A(-x^7)/7 + 2*A(-x^9)/9 + ...).

Original entry on oeis.org

0, 1, -2, -2, 10, 14, -86, -126, 858, 1302, -9378, -14606, 108954, 172698, -1319966, -2119118, 16489594, 26731542, -210887998, -344490170, 2747510514, 4515757426, -36336187630, -60023827438, 486540793914, 807121753178, -6582918170714, -10959656342678, 89860260268098
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 14 2019

Keywords

Examples

			G.f.: A(x) = x - 2*x^2 - 2*x^3 + 10*x^4 + 14*x^5 - 86*x^6 - 126*x^7 + 858*x^8 + 1302*x^9 - 9378*x^10 - 14606*x^11 + ...

Crossrefs

Cf. A000081, A004111, A045648, A049075, A073075, A115593, A306768, A307365, A307366.

Programs

  • Mathematica
    terms = 28; A[] = 0; Do[A[x] = x Exp[Sum[2 A[-x^(2 k - 1)]/(2 k - 1), {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[((1 + x^k)/(1 - x^k))^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 28}]

Formula

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} ((1 + x^n)/(1 - x^n))^((-1)^n*a(n)).
Recurrence: a(n+1) = (2/n) * Sum_{k=1..n} ( Sum_{d|k, k/d odd} (-1)^d*d*a(d) ) * a(n-k+1).
Showing 1-3 of 3 results.

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

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