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 1-7 of 7 results.

A187540 Binomial partial sums of the central Lah numbers.

Original entry on oeis.org

1, 3, 41, 1315, 63825, 4116611, 331127353, 31915763811, 3585520583585, 460054836028675, 66377105303195721, 10637410917472061603, 1874707445757653437681, 360356280811211873453955, 75028021167256736753934425
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187538, A187539, A187542, A187543, A187544, A187545, A187546, A187547, A187548.

Programs

  • Maple
    seq(1+add(binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k=1..n), n=0..20);
  • Mathematica
    Table[1 + Sum[Binomial[n, k]Binomial[2k-1,k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}]
  • Maxima
    makelist(1+sum(binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k,1,n), n,0,12);
  • PARI
    a(n) = 1+sum(k=0,n, binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!) \\ Charles R Greathouse IV, Feb 07 2017

Formula

Formula: a(n) = 1+sum(binomial(n,k)binomial(2k-1,k-1)(2k)!/k!,k=0..n).
Recurrence: for n>=3, a(n) = 1/n*(-2 +(32 - 48*n + 16*n^2)*a(n-3) + (-31 + 63*n - 32*n^2)*a(n-2) + (3 - 14*n + 16*n^2)*a(n-1) )
E.g.f.: exp(x) (1/2 + 1/Pi K(16x) ), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ 16^n*n^(n-1/2)*exp(1/16-n)/sqrt(2*Pi). - Vaclav Kotesovec, Aug 09 2013

A187542 Convolutions of the central Lah numbers (A187535).

Original entry on oeis.org

1, 4, 76, 2544, 123696, 7942080, 635633280, 61009159680, 6831940227840, 874493448514560, 125946241018214400, 20156433977646489600, 3548609812373223628800, 681555522002874494976000, 141810253720479017017344000
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187543, A187544, A187545, A187546, A187547, A187548.

Programs

  • Maple
    a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(a(k)*a(n-k), k=0..n),n=0..12);
  • Mathematica
    a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[a[k]a[n - k], {k, 0, n}], {n, 0, 20}]
  • Maxima
    a(n) := if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(a(k)*a(n-k),k,0,n),n,0,12);

Formula

a(n) = sum(L(k)L(n-k),k=0..n), where L(n) is a central Lah number.
a(n) ~ n! * 16^n / (Pi*n). - Vaclav Kotesovec, Oct 06 2019

A187539 Alternated binomial partial sums of central Lah numbers (A187535).

Original entry on oeis.org

1, 1, 33, 1097, 54209, 3527889, 285356449, 27608615257, 3110179582593, 399896866564001, 57791843384031521, 9273757516482276201, 1636151050649025202753, 314786007405793614831217, 65590496972310741712688289, 14714600180590751334321307769
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187540, A187542, A187543, A187544, A187545, A187546, A187547, A187548.

Programs

  • Maple
    seq((-1)^n+add((-1)^(n-k)*binomial(n,k)*binomial(2*k-1,k-1)*(2*k)!/k!, k=1..n), n=0..20);
  • Mathematica
    Table[(-1)^n + Sum[(-1)^(n-k)Binomial[n,k]Binomial[2k-1,k-1](2k)!/k!, {k, 1, n}], {n, 0, 20}]
  • Maxima
    makelist((-1)^n+sum((-1)^(n-k)*binomial(n,k)*binomial(2*k-1,k-1) *(2*k)!/k!, k,1,n), n,0,12);

Formula

a(n) = 1+sum((-1)^(n-k)*C(n,k)*C(2k-1,k-1)*(2k)!/k!, k=0..n).
Recurrence: n>=3, a(n) = (2*(-1)^n + (32 - 48*n + 16*n^2)*a(n-3) + (33 - 65*n + 32*n^2)*a(n-2) + (5 - 18*n + 16*n^2)*a(n-1))/n
E.g.f.: exp(-x) (1/2 + 1/pi K(16x) ), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ 16^n*n^(n-1/2)/(sqrt(2*Pi)*exp(n+1/16)). - Vaclav Kotesovec, Aug 10 2013

A187544 Stirling transform (of the second kind) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 38, 1310, 66254, 4428782, 368444078, 36691056110, 4256199137774, 563672814445742, 83921091641375918, 13875375391723852910, 2522552600160248918894, 500141581330626431059502, 107400097037199576065830958
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187545, A187546, A187547, A187548.

Programs

  • Maple
    a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(sum(combinat[stirling2](n,k)*a(k), k=0..n),n=0..12);
  • Mathematica
    a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[StirlingS2[n, k]a[k], {k, 0, n}], {n, 0, 20}]
CoefficientList[Series[1/2 + EllipticK[16*(E^x - 1)]/Pi, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 06 2019 *)
  • Maxima
    a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(stirling2(n,k)*a(k),k,0,n),n,0,12);

Formula

a(n) = sum(S(n,k)*L(k),k=0..n), where S(n,k) are the Stirling numbers of the second kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(16(exp(x)-1)) where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (log(17/16))^n). - Vaclav Kotesovec, Oct 06 2019

A187545 Stirling transform (of the first kind) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 38, 1312, 66408, 4442088, 369791064, 36848702784, 4277191653888, 566809715422464, 84441103242634176, 13970100487593468480, 2541362625439551554880, 504185908064687887996800, 108336183242510523080868480
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187546, A187547, A187548.

Programs

  • Maple
    lahc := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(abs(combinat[stirling1](n,k))*lahc(k), k=0..n), n=0..20);
  • Mathematica
    lahc[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[Abs[StirlingS1[n, k]]*lahc[k], {k, 0, n}], {n, 0, 20}]
  • Maxima
    lahc(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(abs(stirling1(n,k))*lahc(k),k,0,n),n,0,12);

Formula

a(n) = sum(s(n,k)*L(k), k=0..n), where s(n,k) are the (signless) Stirling numbers of the first kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(-16*log(1-x)), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (1 - exp(-1/16))^n). - Vaclav Kotesovec, Apr 10 2018

A187546 Stirling transform (of the first kind, with signs) of the central Lah numbers (A187535).

Original entry on oeis.org

1, 2, 34, 1096, 51984, 3262488, 254943384, 23853046656, 2600024557248, 323588157732096, 45276442446814656, 7035574740347812800, 1202158966644148296000, 224022356544364922931840, 45215509996613004825121920
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Links

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187545, A187547, A187548.

Programs

  • Maple
    lahc := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(add(combinat[stirling1](n,k)*lahc(k), k=0..n), n=0..20);
  • Mathematica
    lahc[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[Sum[StirlingS1[n, k]*lahc[k], {k, 0, n}], {n, 0, 20}]
  • Maxima
    lahc(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(sum(stirling1(n,k)*lahc(k),k,0,n),n,0,12);

Formula

a(n) = sum((-1)^(n-k)*s(n,k)*L(k), k=0..n), where s(n,k) are the (signless) Stirling numbers of the first kind and L(n) are the central Lah numbers.
E.g.f.: 1/2 + 1/Pi*K(16*log(1+x)), where K(z) is the elliptic integral of the first kind (defined as in Mathematica).
a(n) ~ n! / (2*Pi*n * (exp(1/16) - 1)^n). - Vaclav Kotesovec, Apr 10 2018

A187547 L(n)H(n+1), product of the central Lah number L(n) and the harmonic number H(n).

Original entry on oeis.org

1, 3, 66, 2500, 134260, 9335088, 796938912, 80671795776, 9446603680800, 1256254443100800, 187033518310129920, 30821040496874234880, 5569495264653352381440, 1095113648992295923200000, 232773183612995427763200000, 53186532693832607435089920000
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Crossrefs

Cf. A187536, A008297, A111596, A187536, A187538, A187539, A187540, A187542, A187543, A187544, A187545, A187546, A187548.

Programs

  • Maple
    a := n -> if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n! fi;
seq(a(n)*sum(1/k,k=1..n+1),n=0..12);
  • Mathematica
    a[n_] := If[n == 0, 1, Binomial[2n - 1, n - 1](2n)!/n!]
Table[a[n]HarmonicNumber[n + 1], {n, 0, 20}]
  • Maxima
    a(n):= if n=0 then 1 else binomial(2*n-1,n-1)*(2*n)!/n!;
makelist(a(n)*sum(1/k,k,1,n+1),n,0,12);

Formula

Recurrence:
(n+3)(n+2)(n+1)a(n+2)-4(2n+3)^2(2n+5)(n+1)a(n+1)+16(2n+3)^2(2n+1)^2(n+2)a(n)-144delta(n,0)=0.
Showing 1-7 of 7 results.

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