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.

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A305560 Expansion of Sum_{k>=0} binomial(k,floor(k/2))*x^k/Product_{j=1..k} (1 - j*x).

Original entry on oeis.org

1, 1, 3, 10, 39, 176, 893, 4985, 30229, 197452, 1379655, 10250087, 80558195, 666916238, 5795111845, 52691973136, 499969246647, 4938724595994, 50679201983653, 539209298355565, 5938139329609621, 67582179415195986, 793755139140445707, 9608367683839952732, 119730171975510540577
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 21 2018

Keywords

Comments

Stirling transform of A001405.

Links

Crossrefs

Cf. A001405, A005773, A064856, A305406.

Programs

  • Maple
    a:= n-> add(binomial(j, floor(j/2))*Stirling2(n, j), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 21 2018
  • Mathematica
    nmax = 24; CoefficientList[Series[Sum[Binomial[k, Floor[k/2]] x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[BesselI[0, 2 (Exp[x] - 1)] + BesselI[1, 2 (Exp[x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 24}]

Formula

E.g.f.: BesselI(0,2*(exp(x) - 1)) + BesselI(1,2*(exp(x) - 1)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(k,floor(k/2)).

A317170 Expansion of e.g.f. exp(exp(x) - 1)*BesselI(1,2*(exp(x) - 1))/(exp(x) - 1).

Original entry on oeis.org

1, 1, 3, 11, 48, 242, 1374, 8619, 58923, 434595, 3431263, 28817120, 256100717, 2397920319, 23567078396, 242343368931, 2600148486462, 29036252825090, 336754427112094, 4048299252733563, 50357053778129599, 647129716643654763, 8579133975080008700, 117178742009906802080, 1646975673395621229201
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 23 2018

Keywords

Comments

Stirling transform of the Motzkin numbers (A001006).

Links

Crossrefs

Cf. A001006, A064856, A086672, A317169.

Programs

  • Maple
    a:=series(exp(exp(x) - 1)*BesselI(1,2*(exp(x) - 1))/(exp(x) - 1), x=0, 26): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] BesselI[1, 2 (Exp[x] - 1)]/(Exp[x] - 1), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4], {k, 0, n}], {n, 0, 24}]

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k)*A001006(k).

A355290 a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling2(n,k) * Catalan(k).

Original entry on oeis.org

1, 1, 1, 0, -3, -2, 23, 17, -333, 86, 6941, -17025, -160267, 1082864, 2273807, -56742606, 152154285, 2293098332, -22007462809, -15179437171, 1671107690083, -10716783889040, -58404948615167, 1439391012463810, -6701658223127029, -88340107011433060
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2022

Keywords

Crossrefs

Cf. A000108, A006531, A064856, A086662, A086672.

Programs

  • Maple
    A355290 := proc(n)
    add((-1)^(n-k)*stirling2(n,k)*A000108(k),k=0..n) ;
end proc:
seq(A355290(n),n=0..70) ; # R. J. Mathar, Mar 13 2023
  • PARI
    a(n) = sum(k=0, n,(-1)^(n-k)*stirling(n, k, 2)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(2*k, k)/(k+1)*x^k/prod(j=1, k, 1+j*x)))

Formula

G.f.: Sum_{k>=0} Catalan(k) * x^k / Product_{j=1..k} (1 + j*x).
Previous Showing 11-13 of 13 results.

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