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.

A187655 Self-convolution of the central Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 15, 194, 3631, 89712, 2764268, 102207394, 4411265695, 217707856946, 12092696127691, 746552539553152, 50708165735187572, 3757864633323765824, 301719332111553586612, 26089939284112306045362, 2417245528055399202851119
Offset: 0

Views

Author

Emanuele Munarini, Mar 12 2011

Keywords

Comments

The sequence of the central Stirling numbers of the second kind is 1, 1, 7, 90, 1701,... with offset 0 (see A007820).

Crossrefs

Cf. A187656.

Programs

  • Maple
    seq( add(combinat[stirling2](2*k,k) *combinat[stirling2](2*(n-k),n-k) ,k=0..n), n=0..12);
  • Mathematica
    Table[Sum[StirlingS2[2k, k]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 16}]
  • Maxima
    makelist(sum(stirling2(2*k,k)*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);

Formula

a(n) = sum_{k=0..n} A048993(2k,k)*A048993(2n-2k,n-k).
a(n) ~ 2^(2*n+1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * (c*(2-c))^n), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 20 2014

A384495 a(n) = Sum_{k=0..n} binomial(n,k)^2 * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)).

Original entry on oeis.org

1, 2, 26, 648, 25094, 1372100, 99827020, 9233563136, 1045169591270, 140259346792380, 21754963505429340, 3823376222328582480, 749784319125445476092, 162122841942093462239368, 38288723630416561023861048, 9801732906198391239249940800, 2702731846233390353066363949830
Offset: 0

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Crossrefs

Cf. A187656, A187658, A384496, A384471.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^2 * Abs[StirlingS1[2*k,k]] * Abs[StirlingS1[2*n-2*k, n-k]], {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(4*n - 1/2) * n^(n - 3/2) * w^(2*n) / ((w-1) * Pi^(3/2) * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2) = 1.7564312086261696769827376166...

A384496 a(n) = Sum_{k=0..n} binomial(n,k)^3 * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)).

Original entry on oeis.org

1, 2, 30, 1044, 68474, 7180900, 1050625720, 196205015216, 44361477901818, 11751610490415828, 3567182462164189140, 1220655384720089761080, 464932034143270233958352, 195108754505934104188716064, 89452431045403310104416682304, 44489455448017524780072427344000
Offset: 0

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Comments

In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)) ~ 2^((m+2)*n + (m-3)/2) * n^(n - (m+1)/2) * w^(2*n) / (sqrt(m-1) * (w-1) * Pi^((m+1)/2) * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2).

Crossrefs

Cf. A187656 (m=0), A187658 (m=1), A384495 (m=2), A384472.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^3 * Abs[StirlingS1[2*k,k]] * Abs[StirlingS1[2*n-2*k,n-k]], {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(5*n - 1/2) * n^(n-2) * w^(2*n) / ((w-1) * Pi^2 * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2) = 1.7564312086261696769827376166...

A384501 a(n) = Sum_{k=0..n} abs(Stirling1(n,k)) * Stirling2(n,n-k).

Original entry on oeis.org

1, 0, 1, 9, 119, 2025, 42510, 1062761, 30854159, 1020615912, 37900765365, 1561459425955, 70682817696436, 3487456195458027, 186281997929231659, 10709829446929099865, 659427284782849503663, 43293574636994934145044, 3019108475859713906967738, 222868205832269470083471366
Offset: 0

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Crossrefs

Cf. A047793, A187655, A187656.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n, k]]*StirlingS2[n, n-k], {k, 0, n}], {n, 0, 20}]
Table[Sum[StirlingS2[n, k]*Abs[StirlingS1[n, n-k]], {k, 0, n}], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} abs(Stirling1(n,n-k)) * Stirling2(n,k).
a(n) ~ c * ((-r - 1/((1-r)*LambertW(exp(1/(r-1))/(r-1)))) / (1 + (1-r)*LambertW(exp(1/(r-1))/(r-1))))^n * n^(n - 1/2) / exp(n), where r = 0.412059483521755003540032983286575579547027818844750... is the root of the equation (1-r)^2 * (1 + LambertW(-1, -exp(-r)*r)/r) = (1-r) + 1/LambertW(exp(1/(r-1))/(r-1)) and c = 0.21367572159147979376975234273...
Showing 1-4 of 4 results.

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