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 1-6 of 6 results.

A345368 a(n) = Sum_{k=0..n} binomial(5*k,k) / (4*k + 1).

Original entry on oeis.org

1, 2, 7, 42, 327, 2857, 26608, 258488, 2588933, 26539288, 277082658, 2936050788, 31494394563, 341325970323, 3731742758203, 41108999917483, 455850863463768, 5084213586320193, 56997201842602368, 641906808539396253, 7258985455500009623, 82393287049581399283
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

Comments

Partial sums of A002294.

Links

Crossrefs

Cf. A002294, A014137, A104859, A345367, A346065, A346647, A346671, A346672.

Programs

  • Mathematica
    Table[Sum[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, binomial(5*k, k)/(4*k+1)); \\ Michel Marcus, Jul 28 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) / (2869 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 28 2021

A346065 a(n) = Sum_{k=0..n} binomial(6*k,k) / (5*k + 1).

Original entry on oeis.org

1, 2, 8, 59, 565, 6046, 68878, 818276, 10021910, 125629220, 1603943486, 20783993414, 272641113110, 3613484662965, 48313969712685, 650888627139801, 8826840286257595, 120398870546499685, 1650711840886884265, 22735860619151166130, 314441081323870331656
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

Comments

Partial sums of A002295.
In general, for m > 1, Sum_{k=0..n} 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 28 2021

Links

Crossrefs

Cf. A002295, A014137, A104859, A345367, A345368, A346648, A346671, A346672.

Programs

  • Mathematica
    Table[Sum[Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]
nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x) + x (1 - x)^5 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • PARI
    a(n) = sum(k=0, n, binomial(6*k, k)/(5*k+1)); \\ Michel Marcus, Jul 28 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^5 * A(x)^6.
a(n) ~ 2^(6*n + 6) * 3^(6*n + 13/2) / (43531 * sqrt(Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Jul 28 2021

A346671 a(n) = Sum_{k=0..n} binomial(7*k,k) / (6*k + 1).

Original entry on oeis.org

1, 2, 9, 79, 898, 11370, 153148, 2150836, 31140511, 461462144, 6964815000, 106691488130, 1654539334220, 25923944408960, 409770113121064, 6526344613981944, 104632592920840659, 1687270854882480906, 27348675382672733281, 445328790513987869681, 7281393330439106226281
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

Comments

Partial sums of A002296.

Links

Crossrefs

Cf. A002296, A014137, A104859, A345367, A345368, A346065, A346649, A346672.

Programs

  • Mathematica
    Table[Sum[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, binomial(7*k, k)/(6*k+1)); \\ Michel Marcus, Jul 28 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) / (776887 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Jul 30 2021

A349364 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^8 / (1 + x).

Original entry on oeis.org

1, 1, 7, 77, 987, 13839, 205513, 3176747, 50578445, 823779286, 13660621282, 229865812134, 3915003083306, 67361559577578, 1169138502393414, 20444573270374050, 359858503314494318, 6370677542063831319, 113359050598950194801, 2026309136822686950087
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Comments

In general, for m > 1, Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (m^m/(m-1)^(m-1) - 1)^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - Vaclav Kotesovec, Nov 17 2021

Links

Crossrefs

Cf. A001006, A007556, A127897, A317133, A346668, A346672 (binomial transform), A349335, A349361, A349362, A349363.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(1+x*A^8/(1+x)-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 15 2021
  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1 + x A[x]^8/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(8*k,k) / (7*k+1).
a(n) = (-1)^(n+1)*F([9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8, 1-n], [9/7, 10/7, 11/7, 12/7, 13/7, 2, 15/7], 8^8/7^7), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
a(n) ~ 15953673^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 17 2021

A345367 a(n) = Sum_{k=0..n} binomial(4*k,k) / (3*k + 1).

Original entry on oeis.org

1, 2, 6, 28, 168, 1137, 8221, 62041, 482773, 3845033, 31188921, 256757719, 2139691083, 18015030073, 153008796673, 1309402039993, 11279339531413, 97724562251137, 851035285261745, 7445189624293545, 65401191955640665, 576639234410182210, 5101317352349364430
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2021

Keywords

Comments

Partial sums of A002293.

Links

Crossrefs

Cf. A002293, A014137, A104859, A345368, A346065, A346646, A346671, A346672.

Programs

  • Mathematica
    Table[Sum[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, binomial(4*k, k)/(3*k+1)); \\ Michel Marcus, Jul 28 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) / (229 * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jul 28 2021
D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) +(-283*n^3+384*n^2-173*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

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 1-6 of 6 results.

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