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.

A346680 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*k,k) / (3*k + 1).

Original entry on oeis.org

1, 0, 4, 18, 122, 847, 6237, 47583, 373149, 2989111, 24354777, 201214021, 1681719343, 14193619647, 120800146953, 1035593096367, 8934344395053, 77510878324671, 675799844685937, 5918354494345863, 52037647837001257, 459200394617540288, 4065477723321641932
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 29 2021

Keywords

Links

Crossrefs

Cf. A002293, A032357, A188678, A345367, A346664.
Cf. A346681, A346682, A346683, A346684.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^3 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(4*k, k)/(3*k + 1)); \\ Michel Marcus, Jul 29 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^3 * A(x)^4.
a(n) ~ 2^(8*n + 17/2) / (283 * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) -(n-1)*(229*n^2-155*n+24)*a(n-1) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Aug 05 2021

A346681 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(5*k,k) / (4*k + 1).

Original entry on oeis.org

1, 0, 5, 30, 255, 2275, 21476, 210404, 2120041, 21830314, 228713056, 2430255074, 26128088701, 283703487059, 3106713300821, 34270543858459, 380471319687826, 4247891403168599, 47665096853113576, 537244509843680309, 6079834137116933061, 69054467456964456599
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 29 2021

Keywords

Links

Crossrefs

Cf. A002294, A032357, A188678, A345368, A346665.
Cf. A346680, A346682, A346683, A346684.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^4 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(5*k, k)/(4*k + 1)); \\ Michel Marcus, Jul 29 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^4 * A(x)^5.
a(n) ~ 5^(5*n + 11/2) / (3381 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 30 2021

A346683 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(7*k,k) / (6*k + 1).

Original entry on oeis.org

1, 0, 7, 63, 756, 9716, 132062, 1865626, 27124049, 403197584, 6100155272, 93626517858, 1454221328232, 22815183746508, 361030984965596, 5755543515895284, 92350704790963431, 1490287557170676816, 24171116970619575559, 393808998160695560841, 6442255541764422795759
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 29 2021

Keywords

Links

Crossrefs

Cf. A002296, A032357, A188678, A346667, A346671.
Cf. A346680, A346681, A346682, A346684.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^6 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(7*k, k)/(6*k + 1)); \\ Michel Marcus, Jul 29 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^6 * A(x)^7.
a(n) ~ 7^(7*n + 15/2) / (870199 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A346684 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(8*k,k) / (7*k + 1).

Original entry on oeis.org

1, 0, 8, 84, 1156, 17122, 268262, 4370086, 73281938, 1256608767, 21933420953, 388400019583, 6960642974905, 126008367913375, 2300862338502425, 42326714610861679, 783717720798538121, 14594469249932149279, 273161824453612674593, 5135931850101477641707
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 29 2021

Keywords

Comments

In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(m*k,k) / ((m-1)*k + 1) ~ m^(m*(n+1) + 1/2) / (sqrt(2*Pi) * (m^m + (m-1)^(m-1)) * n^(3/2) * (m-1)^((m-1)*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

Links

Crossrefs

Cf. A007556, A032357, A188678, A346668, A346672.
Cf. A346680, A346681, A346682, A346683.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]
nmax = 19; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^7 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(8*k, k)/(7*k + 1)); \\ Michel Marcus, Jul 29 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^7 * A(x)^8.
a(n) ~ 2^(24*n + 25) / (17600759 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021
Showing 1-4 of 4 results.

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