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 1-10 of 12 results. Next

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).

A307366 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, 0, 0, 1, -2, -1, 3, 3, -8, -5, 17, 15, -47, -35, 118, 91, -311, -240, 839, 660, -2314, -1809, 6417, 5035, -18002, -14177, 51016, 40322, -145784, -115402, 419197, 332457, -1212617, -963586, 3526976, 2807301, -10307097, -8215194, 30246994, 24139050, -89101081
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 05 2019

Keywords

Examples

			G.f.: A(x) = x - x^2 + x^5 - 2*x^6 - x^7 + 3*x^8 + 3*x^9 - 8*x^10 - 5*x^11 + 17*x^12 + ...

Crossrefs

Cf. A000081, A004111, A045648, A049075, A307365.

Programs

  • Mathematica
    terms = 42; A[] = 0; Do[A[x] = x Exp[Sum[(-1)^(k + 1) 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 + x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 42}]

Formula

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

A345234 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, 2, 3, 5, 9, 17, 31, 58, 112, 218, 427, 844, 1683, 3381, 6824, 13842, 28226, 57796, 118762, 244874, 506515, 1050688, 2185095, 4555217, 9517423, 19926174, 41798031, 87833877, 184881588, 389765182, 822901122, 1739763655, 3682955618, 7806103024, 16564348106, 35187631009
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A007560, A007562, A049075, A345235.

Programs

  • Mathematica
    nmax = 38; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[(-1)^(k + 1) 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[(-1)^(k + 1) Sum[(-1)^(k/d + d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 38}]

Formula

G.f.: x + x^2 * Product_{n>=1} (1 + (-x)^n)^((-1)^n*a(n)).
a(n+2) = (1/n) * Sum_{k=1..n} (-1)^(k+1) * ( Sum_{d|k} (-1)^(k/d+d) * d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 2.21094707842288180828190718521597733363607957468229824761... and c = 0.664585976397397791197984310778764361056468131968... - Vaclav Kotesovec, Jun 19 2021

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).

A308245 G.f.: x * Product_{k>=1} 1/(1 - a(k)*(-x)^k)^((-1)^k).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 34, 76, 168, 368, 838, 1964, 4544, 10464, 24658, 58984, 140072, 331456, 795834, 1932228, 4665304, 11227280, 27305882, 66953236, 163418448, 397826496, 976658846, 2412163316, 5935476672, 14576596320, 36023097266, 89458468968
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2019

Keywords

Crossrefs

Cf. A032305, A045648, A049075, A093637, A308246.

Programs

  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - a[k] (-x)^k)^((-1)^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 33}]
a[n_] := a[n] = Sum[Sum[(-1)^(k + d) d a[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]/(n - 1); a[1] = 1; Table[a[n], {n, 1, 33}]

Formula

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k+d)*d*a(d)^(k/d) ) * a(n-k+1).

A308246 G.f.: x * Product_{k>=1} (1 + a(k)*(-x)^k)^((-1)^k).

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 44, 104, 246, 620, 1600, 4082, 10436, 27360, 73046, 193296, 509984, 1371214, 3727792, 10065872, 27145058, 74142688, 204005440, 558475342, 1527058912, 4213709856, 11694035010, 32331790700, 89266126856, 248240818282, 693599213260
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2019

Keywords

Crossrefs

Cf. A032305, A045648, A049075, A093637, A308245.

Programs

  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[x Product[(1 + a[k] (-x)^k)^((-1)^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 31}]
a[n_] := a[n] = Sum[Sum[(-1)^(k/d + k + d + 1) d a[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]/(n - 1); a[1] = 1; Table[a[n], {n, 1, 31}]

Formula

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+k+d+1)*d*a(d)^(k/d) ) * a(n-k+1).

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).

A345884 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} (-1)^k * A(x^k) / k).

Original entry on oeis.org

1, -2, 7, -26, 103, -442, 1982, -9122, 42985, -206526, 1007322, -4974066, 24819268, -124949782, 633882799, -3237261340, 16629986395, -85873762466, 445491479309, -2320717519612, 12134813554225, -63667883444468, 335083404759136, -1768545061282712, 9358571746569760
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 28 2021

Keywords

Crossrefs

Cf. A000151, A005753, A049075, A345885.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1, 2*add(a(n-k)*add(d*a(d)
       *(-1)^(k/d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 28 2021
  • Mathematica
    nmax = 25; A[] = 0; Do[A[x] = x Exp[2 Sum[(-1)^k A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[n_] := a[n] = (2/(n - 1)) Sum[Sum[(-1)^(k/d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 25}]

Formula

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

A345885 G.f. A(x) satisfies: A(x) = x * exp(3 * Sum_{k>=1} (-1)^k * A(x^k) / k).

Original entry on oeis.org

1, -3, 15, -82, 486, -3090, 20497, -140010, 979131, -6976603, 50461716, -369533691, 2734423934, -20414010219, 153571115619, -1163003999342, 8859172575069, -67835214598017, 521824159637718, -4030828937892966, 31252886542570119, -243142210911325273, 1897466281615297698
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 28 2021

Keywords

Crossrefs

Cf. A006964, A049075, A052757, A345884.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1, 3*add(a(n-k)*add(d*a(d)
      *(-1)^(k/d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=1..23); # Alois P. Heinz, Jun 28 2021
  • Mathematica
    nmax = 23; A[] = 0; Do[A[x] = x Exp[3 Sum[(-1)^k A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[n_] := a[n] = (3/(n - 1)) Sum[Sum[(-1)^(k/d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 23}]

Formula

G.f.: x / Product_{n>=1} (1 + x^n)^(3*a(n)).
a(n+1) = (3/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * a(d) ) * a(n-k+1).
Showing 1-10 of 12 results. Next

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