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.

A351767 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x)^3.

Original entry on oeis.org

1, 4, 25, 214, 2293, 29176, 427189, 7049890, 129178249, 2597880268, 56815155121, 1341068392654, 33951269718205, 917020113259264, 26305693331946253, 798293630021120986, 25540244079135784849, 858854698277997113620, 30274382852181639467209
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2023

Keywords

Links

Crossrefs

Column k=3 of A361616.
Cf. A052852, A361599.

Programs

  • Mathematica
    Table[n!*Sum[Binomial[n + 2*k + 2, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)^3))
  • PARI
    a(n) = n! * sum(k=0, n, binomial(n+2*k+2,n-k)/k!); \\ Winston de Greef, Mar 18 2023

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+2*k+2,n-k)/k! = Sum_{k=0..n} (n+2*k+2)!/(3*k+2)! * binomial(n,k).
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) = 4*n*a(n-1) - (n-1)*(6*n - 5)*a(n-2) + (n-2)*(n-1)*(4*n - 3)*a(n-3) - (n-3)*(n-2)*(n-1)^2*a(n-4).
a(n) ~ exp(-1/27 - 3^(-5/4)*n^(1/4)/8 + sqrt(n/3)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n + 5/8) / (2 * 3^(5/8)) * (1 + 91837/69120 * 3^(1/4)/n^(1/4)). (End)

A361600 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j,k*j)/j!.

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 1, 2, 7, 16, 1, 2, 9, 34, 65, 1, 2, 11, 58, 209, 326, 1, 2, 13, 88, 473, 1546, 1957, 1, 2, 15, 124, 881, 4626, 13327, 13700, 1, 2, 17, 166, 1457, 10526, 52537, 130922, 109601, 1, 2, 19, 214, 2225, 20326, 145867, 677594, 1441729, 986410
Offset: 0

Views

Author

Seiichi Manyama, Mar 17 2023

Keywords

Examples

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    2,    2,    2,     2,     2,     2, ...
    5,    7,    9,    11,    13,    15, ...
   16,   34,   58,    88,   124,   166, ...
   65,  209,  473,   881,  1457,  2225, ...
  326, 1546, 4626, 10526, 20326, 35226, ...

Links

Crossrefs

Columns k=0..3 give A000522, A002720, A361598, A361599.
Main diagonal gives A361607.
Cf. A293012.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n, binomial(n+(k-1)*j, k*j)/j!);

Formula

E.g.f. of column k: exp( x/(1 - x)^k ) / (1-x).
T(n,k) = Sum_{j=0..n} (n+(k-1)*j)!/(k*j)! * binomial(n,j).

A361626 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x)^2.

Original entry on oeis.org

1, 3, 17, 139, 1437, 17711, 252133, 4059567, 72779129, 1435276027, 30836352441, 716101686323, 17858449006357, 475653606922599, 13467411746316557, 403708230041927191, 12767545998797849073, 424670548932688771187, 14814998283177691422049
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2023

Keywords

Links

Crossrefs

Cf. A091695, A351767, A361599.
Cf. A000262, A001339, A343884.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)^2))
  • PARI
    a(n)=n! * sum(k=0, n, binomial(n+2*k+1,n-k)/k!) \\ Winston de Greef, Mar 18 2023

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+2*k+1,n-k)/k! = Sum_{k=0..n} (n+2*k+1)!/(3*k+1)! * binomial(n,k).
a(n) ~ 3^(5/8) * exp(-1/27 - 3^(3/4)*n^(1/4)/72 + sqrt(3*n)/6 + 4*3^(-3/4)*n^(3/4) - n) * n^(n + 3/8) / 6 * (1 + 63037 * 3^(1/4)/(69120 * n^(1/4))). - Vaclav Kotesovec, Mar 29 2023

A375172 Expansion of e.g.f. exp( x^2/(1-x)^3 ) / (1-x).

Original entry on oeis.org

1, 1, 4, 30, 276, 2940, 36120, 507360, 8032080, 141235920, 2725107840, 57151211040, 1293129351360, 31376876731200, 812303844992640, 22338850742208000, 650081402588217600, 19951131574037664000, 643805564147435289600, 21785365857810973017600
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2024

Keywords

Crossrefs

Cf. A375224, A375225.
Cf. A361595, A361599.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x^2/(1-x)^3)/(1-x)))
  • PARI
    a(n) = n!*sum(k=0, n\2, binomial(n+k, n-2*k)/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n+k,n-2*k)/k!.

A373771 Expansion of e.g.f. exp(x^2 / (2 * (1 - x)^3)) / (1 - x).

Original entry on oeis.org

1, 1, 3, 18, 147, 1425, 15855, 200130, 2838465, 44767485, 777046095, 14705245170, 301014595035, 6621102973485, 155640761791515, 3891902825660850, 103115436832433025, 2884715829245475225, 84950805438277854075, 2626194012669689512050
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2024

Keywords

Crossrefs

Cf. A130905, A361596, A373770.
Cf. A361597, A361599.
Cf. A335345.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, binomial(n+k, n-2*k)/(2^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n+k,n-2*k)/(2^k * k!).

A377599 E.g.f. satisfies A(x) = exp( x * A(x) / (1-x)^2 ) / (1-x).

Original entry on oeis.org

1, 2, 13, 145, 2277, 46461, 1172713, 35374697, 1243296169, 49940748073, 2258238723021, 113567169318285, 6289161888870061, 380364426242671469, 24948313525570134001, 1764095427822803465521, 133782341347522663175889, 10832097536377585282160337, 932693691617428946786304661
Offset: 0

Views

Author

Seiichi Manyama, Nov 14 2024

Keywords

Links

Crossrefs

Cf. A367789, A377608, A377811.
Cf. A361599.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^3))/(1-x)))
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+2*k, n-k)/k!);

Formula

E.g.f.: exp( -LambertW(-x/(1-x)^3) )/(1-x).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+2*k,n-k)/k!.
Showing 1-6 of 6 results.

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