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 1-5 of 5 results.

A217484 Partial sums of the numbers in sequence A080253.

Original entry on oeis.org

1, 4, 21, 168, 1865, 26348, 450205, 9011152, 206624529, 5338349652, 153408637349, 4853054571896, 167576795780953, 6271355892192316, 252836327218276653, 10924378168890333600, 503589353964709474337, 24669610145575233317540
Offset: 0

Views

Author

Emanuele Munarini, Oct 04 2012

Keywords

Links

Crossrefs

Cf. A080253, A000670, A217483, A217485, A217486, A217487, A217488.

Programs

  • Mathematica
    t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[c[k], {k,0,n}], {n,0,100}]
  • Maxima
    t(n):=sum(stirling2(n,k)*k!,k,0,n);
c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
makelist(sum(c(k),k,0,n),n,0,10);

Formula

a(n) = sum(c(k),k=0..n), where c(n) = A080253(n).
E.g.f.: exp (x)/(2-exp(2*x)) + x*exp (x)/2 + (1/4)*exp(x)*log(1/(2-exp(2*x))). - corrected by Vaclav Kotesovec, Jan 02 2013
a(n) ~ n! * 2^(n-1/2)/(log(2))^(n+1). - Vaclav Kotesovec, Jan 02 2013

A217486 Binomial convolution of the numbers in sequence A080253.

Original entry on oeis.org

1, 6, 52, 600, 8656, 149856, 3026752, 69866880, 1814338816, 52350752256, 1661575754752, 57531530434560, 2158011794968576, 87173881613869056, 3772959800981143552, 174183372619165040640, 8543978588021450407936, 443748799382401230176256
Offset: 0

Views

Author

Emanuele Munarini, Oct 04 2012

Keywords

Links

Crossrefs

Cf. A080253, A000670, A217483, A217484, A217485, A217487, A217488.

Programs

  • Mathematica
    t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[Binomial[n,k]c[k]c[n-k], {k,0,n}], {n,0,100}]; Table[2^n t[n+1], {n,0,100}]
With[{nn=20},CoefficientList[Series[Exp[2x]/(2-Exp[2x])^2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 09 2017 *)
  • Maxima
    t(n):=sum(stirling2(n,k)*k!,k,0,n);
c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
makelist(sum(binomial(n,k)*c(k)*c(n-k),k,0,n),n,0,10);
makelist(2^n*t(n+1),n,0,40);
  • Sage
    def A217486(n):
    return 2^n*add(add((-1)^(j-i)*binomial(j,i)*i^(n+1) for i in range(n+2)) for j in range(n+2))
[A217486(n) for n in range(18)] # Peter Luschny, Jul 22 2014

Formula

a(n) = sum(binomial(n,k)*c(k)*c(n.k),k=0..n), where c(n) = A080253(n).
a(n) = 2^n*t(n+1), where t(n) = ordered Bell numbers (A000670).
E.g.f. exp(2*x)/(2-exp(2*x))^2.
G.f.: 1/G(0) where G(k) = 1 - x*3*(2*k+2) + x^2*(k+1)*(k+2)*(1-3^2)/G(k+1) ; (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Jan 11 2013.
a(n) ~ n!*n*2^(n-1)/(log(2))^(n+2). - Vaclav Kotesovec, Aug 11 2013

A217483 Alternating sums of the numbers in sequence A080253.

Original entry on oeis.org

1, 2, 15, 132, 1565, 22918, 400939, 8160008, 189453369, 4942271754, 143128015943, 4556517918604, 158167223290453, 5945611873120910, 240619359452963427, 10430922482219093520, 482234053313600047217, 23683786738296923795986
Offset: 0

Views

Author

Emanuele Munarini, Oct 04 2012

Keywords

Crossrefs

Cf. A080253, A000670, A217484, A217485, A217486, A217487, A217488.

Programs

  • Mathematica
    t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[(-1)^(n-k)c[k],{k,0,n}], {n, 0, 100}]
nmax = 20; CoefficientList[Series[E^x/(2 - E^(2*x)) + Log[2 - E^(2*x)] / (2*E^x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 27 2017 *)
  • Maxima
    t(n):=sum(stirling2(n,k)*k!,k,0,n);
c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
makelist(sum((-1)^(n-k)*c(k),k,0,n),n,0,10);

Formula

a(n) = sum((-1)^(n-k)*c(k),k=0..n), where c(n) = A080253(n).
E.g.f.: exp(x)/(2-exp(2*x)) - (1/2)*exp(-x)*log(1/(2-exp(2*x))). - corrected by Vaclav Kotesovec, Nov 27 2017
a(n) ~ n! * 2^(n - 1/2) / (log(2))^(n+1). - Vaclav Kotesovec, Nov 27 2017

A217485 Convolution of the numbers in sequence A080253.

Original entry on oeis.org

1, 6, 43, 396, 4565, 64146, 1073919, 20996376, 471081385, 11947911966, 338204687315, 10570101018276, 361458024882045, 13421571912745386, 537661560385125031, 23108777539028187696, 1060571767117824260945, 51760585513634983767606
Offset: 0

Views

Author

Emanuele Munarini, Oct 04 2012

Keywords

Crossrefs

Cf. A080253, A000670, A217483, A217484, A217486, A217487, A217488.

Programs

  • Mathematica
    t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[c[k]c[n-k], {k,0,n}], {n,0,100}]
  • Maxima
    t(n):=sum(stirling2(n,k)*k!,k,0,n);
c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
makelist(sum(c(k)*c(n-k),k,0,n),n,0,40);

Formula

a(n) = sum(c(k)*c(n.k),k=0..n), where c(n) = A080253(n).
a(n) ~ n! * 2^(n + 1/2) / (log(2))^(n+1). - Vaclav Kotesovec, Nov 27 2017

A217487 Partial sums of the squares of the numbers in sequence A080253.

Original entry on oeis.org

1, 10, 299, 21908, 2901717, 602319006, 180257075455, 73470070612264, 39124516839956393, 26373727254869321522, 21951183825927218885331, 22108623093930072226980540, 26501124576166085360405809789, 37282620382364481062065321327558
Offset: 0

Views

Author

Emanuele Munarini, Oct 04 2012

Keywords

Crossrefs

Cf. A080253, A000670, A217483, A217484, A217485, A217486, A217488.

Programs

  • Mathematica
    t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[c[k]^2, {k,0,n}], {n,0,100}]
  • Maxima
    t(n):=sum(stirling2(n,k)*k!,k,0,n);
c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
makelist(sum(c(k)^2,k,0,n),n,0,40);

Formula

a(n) = sum(c(k)^2,k=0..n), where c(n) = A080253(n).
a(n) ~ (n!)^2 * 2^(2*n-1) / (log(2))^(2*n + 2). - Vaclav Kotesovec, Nov 27 2017
Showing 1-5 of 5 results.

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