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.

A349581 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^2 * A(x)^4.

Original entry on oeis.org

1, 3, 12, 66, 460, 3681, 31848, 289176, 2714044, 26103468, 255876048, 2546717454, 25666830724, 261407935366, 2686191839232, 27815564456544, 289960011573212, 3040424427011492, 32046741183678288, 339345854532800136, 3608307717155678256, 38511520730570169033
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 22 2021

Keywords

Comments

Second binomial transform of A002293.

Crossrefs

Cf. A002293, A064613, A346646, A346762, A349582, A349584, A349590, A349591.

Programs

  • Mathematica
    nmax = 21; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^2 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[4 k, k] 2^(n - k)/(3 k + 1), {k, 0, n}], {n, 0, 21}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(4*k,k)*2^(n-k)/(3*k+1)); \\ Michel Marcus, Nov 23 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k) * 2^(n-k) / (3*k+1).
a(n) = 2^n*F([1/4, 1/2, 3/4, -n], [2/3, 1, 4/3], -2^7/3^3), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 2^(n - 10) * 155^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Nov 26 2021

A349582 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^3 * A(x)^5.

Original entry on oeis.org

1, 3, 13, 85, 733, 7292, 78267, 880250, 10226237, 121713373, 1476272394, 18180126906, 226704989103, 2856790765238, 36321840773980, 465362291912648, 6002272018481901, 77873186277771107, 1015583616140910999, 13306207249869273003, 175064043975233981626
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 22 2021

Keywords

Comments

Second binomial transform of A002294.

Crossrefs

Cf. A002294, A064613, A346647, A346762, A349581, A349584, A349590, A349591.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^3 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[5 k, k] 2^(n - k)/(4 k + 1), {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(5*k,k)*2^(n-k)/(4*k+1)); \\ Michel Marcus, Nov 23 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*k,k) * 2^(n-k) / (4*k+1).
a(n) = 2^n*F([1/5, 2/5, 3/5, 4/5, -n], [1/2, 3/4, 1, 5/4], -5^5/2^9), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 3637^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 26 2021

A349584 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^4 * A(x)^6.

Original entry on oeis.org

1, 3, 14, 107, 1106, 13173, 168820, 2264298, 31356818, 444803666, 6429510234, 94356870748, 1402149248128, 21055387206719, 319007902203196, 4870481885025752, 74858763620576738, 1157339247553310574, 17985974981514604660, 280813589679135551721
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 22 2021

Keywords

Comments

Second binomial transform of A002295.

Crossrefs

Cf. A002295, A064613, A346648, A346762, A349581, A349582, A349590, A349591.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^4 A[x]^6 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[6 k, k] 2^(n - k)/(5 k + 1), {k, 0, n}], {n, 0, 19}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(6*k,k)*2^(n-k)/(5*k+1)); \\ Michel Marcus, Nov 23 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(6*k,k) * 2^(n-k) / (5*k+1).
a(n) = 2^n*F([1/6, 1/3, 1/2, 2/3, 5/6, -n], [2/5, 3/5, 4/5, 1, 6/5], -3^6*(2/5)^5), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 2^(n - 15/2) * 26453^(n + 3/2) / (6561 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 26 2021

A349590 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^5 * A(x)^7.

Original entry on oeis.org

1, 3, 15, 132, 1595, 22134, 329718, 5136028, 82579819, 1359902823, 22818697128, 388728802702, 6705324823466, 116878939752376, 2055505806198352, 36427660285955808, 649894104351874395, 11662729497015257677, 210383830525447606431, 3812719304673511150854
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 22 2021

Keywords

Comments

Second binomial transform of A002296.

Crossrefs

Cf. A002296, A064613, A346649, A346762, A349581, A349582, A349584, A349591.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^5 A[x]^7 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[7 k, k] 2^(n - k)/(6 k + 1), {k, 0, n}], {n, 0, 19}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(7*k,k)*2^(n-k)/(6*k+1)); \\ Michel Marcus, Nov 23 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(7*k,k) * 2^(n-k) / (6*k+1).
a(n) = 2^n*F([1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -n], [1/3, 1/2, 2/3, 5/6, 1, 7/6], -7^7/(2^7*3^6)), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 916855^(n + 3/2) / (282475249 * sqrt(Pi) * n^(3/2) * 3^(6*n + 3/2) * 4^(3*n + 1)). - Vaclav Kotesovec, Nov 26 2021

A349591 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x) + x * (1 - 2*x)^6 * A(x)^8.

Original entry on oeis.org

1, 3, 16, 160, 2216, 35110, 596016, 10573748, 193586424, 3629709697, 69342483276, 1344897261828, 26411276859800, 524117511080056, 10493756451964088, 211719733855698808, 4300202981875132408, 87854045612854431128, 1804215079309443709632
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 22 2021

Keywords

Comments

Second binomial transform of A007556.

Crossrefs

Cf. A007556, A064613, A346650, A346762, A349581, A349582, A349584, A349590.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = 1/(1 - 2 x) + x (1 - 2 x)^6 A[x]^8 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[8 k, k] 2^(n - k)/(7 k + 1), {k, 0, n}], {n, 0, 18}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(8*k,k)*2^(n-k)/(7*k+1)); \\ Michel Marcus, Nov 23 2021

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(8*k,k) * 2^(n-k) / (7*k+1).
a(n) = 2^n*F([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -2^23/7^7), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 22 2021
a(n) ~ 2^(n - 67/2) * 9212151^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 26 2021

A346763 G.f. A(x) satisfies: A(x) = 1 / (1 - 3*x) + x * (1 - 3*x) * A(x)^3.

Original entry on oeis.org

1, 4, 18, 93, 550, 3636, 26079, 197931, 1562382, 12685116, 105187512, 886700898, 7574331987, 65413265014, 570155069547, 5008957733472, 44306834969838, 394269180748272, 3527034255411864, 31700659283908242, 286124960854479888, 2592334353741781752, 23567790327842864046
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 02 2021

Keywords

Comments

Third binomial transform of A001764.

Crossrefs

Cf. A001764, A104455, A188687, A346762.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1/(1 - 3 x) + x (1 - 3 x) A[x]^3 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n, k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
Table[3^n HypergeometricPFQ[{1/3, 2/3, -n}, {1, 3/2}, -9/4], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*k,k) * 3^(n-k) / (2*k + 1).
a(n) ~ 3^(n - 5/2) * 13^(n + 3/2) / (sqrt(Pi) * n^(3/2) * 2^(2*(n+1))). - Vaclav Kotesovec, Nov 26 2021
Showing 1-6 of 6 results.

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