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 1-10 of 10 results.

A229234 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n*k*x).

Original entry on oeis.org

1, 1, 3, 19, 189, 2671, 50253, 1203679, 35548509, 1263153631, 52973381853, 2581493517439, 144317666200029, 9156299509121311, 653254398215833053, 51995430120141924799, 4585316010326597014749, 445304380297565009962591, 47368550666889620425580253, 5492643630110295899167573759
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 189*x^4 + 2671*x^5 + 50253*x^6 +...
where
A(x) = 1 + x/(1-x) + 2!*x^2/((1-2*1*x)*(1-2*2*x)) + 3!*x^3/((1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4!*x^4/((1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 189*x^4/4! + 2671*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2/2^2 + (exp(3*x)-1)^3/3^3 + (exp(4*x)-1)^4/4^4 + (exp(5*x)-1)^5/5^5 +...

Links

Crossrefs

Cf. A229233, A220181, A108459, A122399.
Cf. A229258, A229259, A229260, A229261.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[k^(n-k) * k! * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1-m*k*x +x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,(exp(m*x+x*O(x^n))-1)^m/m^m),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0, n, k^(n-k) * k! * stirling(n, k, 2))}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} k^(n-k) * k! * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n*x) - 1)^n / n^n.

A229260 O.g.f.: Sum_{n>=0} n! * n^(2*n) * x^n / Product_{k=1..n} (1 - n^2*k*x).

Original entry on oeis.org

1, 1, 33, 4759, 1812645, 1432421311, 2033196095973, 4707913008727279, 16598602853910799125, 84603008117292025844671, 598699398082553327852353413, 5694542805400507375406964870799, 70891082687197321771955383523878005, 1129717853570486718325946169950885995231
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 33*x^2 + 4759*x^3 + 1812645*x^4 + 1432421311*x^5 +...
where
A(x) = 1 + x/(1-x) + 2!*2^4*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*3^6*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*4^8*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 33*x^2/2! + 4759*x^3/3! + 1812645*x^4/4! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2 + (exp(9*x)-1)^3 + (exp(16*x)-1)^4 + (exp(25*x)-1)^5 + (exp(36*x)-1)^6 + (exp(49*x)-1)^7 +...

Links

Crossrefs

Cf. A229257, A229258, A229259, A229261, A229233, A229234, A220181, A122399.
Cf. A187755.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[k^(2*n) * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,m!*m^(2*m)*x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=sum(k=0, n, k^(2*n) * k! * Stirling2(n, k))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} k^(2*n) * k! * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n.
a(n) ~ c * d^n * (n!)^3 / n, where d = r^3*(1+exp(2/r)) = 7.8512435106631367719817991716164612615296980032514..., r = 0.94520217245242431308104743874492469552738... is the root of the equation (1+exp(-2/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.142680262107781025906560380273234930916319644... . - Vaclav Kotesovec, May 08 2014

A229261 O.g.f.: Sum_{n>=0} n^(2*n) * x^n / Product_{k=1..n} (1 - n^2*k*x).

Original entry on oeis.org

1, 1, 17, 922, 106695, 21742971, 6977367418, 3273755821827, 2129976884025085, 1846718792259030760, 2068516760060790309349, 2919795339100534415091143, 5088912154987483773753872912, 10766599670032172748225017763021, 27254500086981764567988714050736205
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 17*x^2 + 922*x^3 + 106695*x^4 + 21742971*x^5 +...
where
A(x) = 1 + x/(1-x) + 2^4*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3^6*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4^8*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 17*x^2/2! + 922*x^3/3! + 106695*x^4/4! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/2! + (exp(9*x)-1)^3/3! + (exp(16*x)-1)^4/4! + (exp(25*x)-1)^5/5! + (exp(36*x)-1)^6/6! +...

Links

Crossrefs

Cf. A229257, A229258, A229259, A229260, A229233, A229234, A220181, A122399.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[k^(2*n) * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,m^(2*m)*x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/m!),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0, n, k^(2*n) * stirling(n, k, 2))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} k^(2*n) * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / n!.

A229258 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n^2*k*x).

Original entry on oeis.org

1, 1, 3, 31, 573, 18031, 854613, 57433951, 5242645173, 625589806831, 95051257799973, 17976303383444671, 4153215615930529173, 1154304694449774708751, 380809177225169291456133, 147420687475847638142996191, 66303807316628093952943203573
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 3*x^2 + 31*x^3 + 573*x^4 + 18031*x^5 + 854613*x^6 +...
where
A(x) = 1 + x/(1-x) + 2!*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 573*x^4/4! + 18031*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/4^2 + (exp(9*x)-1)^3/9^3 + (exp(16*x)-1)^4/16^4 + (exp(25*x)-1)^5/25^5 +...

Links

Crossrefs

Cf. A229257, A229259, A229260, A229261; A229233, A229234, A220181, A122399.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[(k^2)^(n-k) * k! * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/m^(2*m)),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0, n, (k^2)^(n-k) * k! * stirling(n, k, 2))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} (k^2)^(n-k) * k! * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / n^(2*n).

A229259 O.g.f.: Sum_{n>=0} n! * n^n * x^n / Product_{k=1..n} (1 - n^2*k*x).

Original entry on oeis.org

1, 1, 9, 259, 15789, 1693771, 287145789, 71487432619, 24798142070109, 11518873418467051, 6945333793188487869, 5301472723402989073579, 5018547949600497090304029, 5790959348524892656227425131, 8026963462960378548022418765949, 13197920271743736945902641688868139
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 9*x^2 + 259*x^3 + 15789*x^4 + 1693771*x^5 +...
where
A(x) = 1 + x/(1-x) + 2!*2^2*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*3^3*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*4^4*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 9*x^2/2! + 259*x^3/3! + 15789*x^4/4! + 1693771*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/2^2 + (exp(9*x)-1)^3/3^3 + (exp(16*x)-1)^4/4^4 + (exp(25*x)-1)^5/5^5 +...

Links

Crossrefs

Cf. A229257, A229258, A229260, A229261, A229233, A229234, A220181, A122399.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[k^(2*n-k) * k! * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,m!*m^m*x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/m^m),n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0, n, k^(2*n-k) * k! * stirling(n, k, 2))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} k^(2*n-k) * k! * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / n^n.

A229257 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n^2*k*x).

Original entry on oeis.org

1, 1, 2, 14, 168, 3147, 90563, 3561231, 185790622, 12599020184, 1071164190670, 111813313594259, 14140296360430353, 2132273568722682621, 378197030144360862958, 78127192632748956075174, 18627308660113953164384812, 5081218748742336002185874439, 1574128413278644602881499193579
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2013

Keywords

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 168*x^4 + 3147*x^5 + 90563*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/((1-2^2*1*x)*(1-2^2*2*x)) + x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 168*x^4/4! + 3147*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/(2!*4^2) + (exp(9*x)-1)^3/(3!*9^3) + (exp(16*x)-1)^4/(4!*16^4) + (exp(25*x)-1)^5/(5!*25^5) +...

Links

Crossrefs

Cf. A229258, A229259, A229260, A229261; A229233, A229234, A220181, A122399.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[(k^2)^(n-k) * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/(m!*m^(2*m))),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0, n, (k^2)^(n-k) * stirling(n, k, 2))}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} (k^2)^(n-k) * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / (n! * n^(2*n)).

A350726 a(n) = Sum_{k=0..n} k^(n-k) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 0, -3, 21, -100, -525, 33026, -860503, 16304464, -100885935, -12798492630, 1037135603845, -55556702499792, 2207903148318777, -31916679640973750, -6164889702150516015, 983802138243128355456, -100629406324320358067423
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2022

Keywords

Crossrefs

Cf. A229233, A305819, A320082, A350725.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k^(n - k) * StirlingS1[n, k], {k, 1, n}]; Array[a, 19, 0] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = sum(k=0, n, k^(n-k)*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k*x)^k/(k!*k^k))))

Formula

E.g.f.: Sum_{k>=0} log(1 + k*x)^k / (k! * k^k).

A351181 a(n) = Sum_{k=0..n} k^(k+n) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 17, 826, 79107, 12553011, 2979141058, 988163147091, 436562014218313, 247800100563125728, 175732698005376526429, 152264214647249387402567, 158273183995563848011907696, 194391589002961482387840145341
Offset: 0

Views

Author

Seiichi Manyama, Feb 04 2022

Keywords

Crossrefs

Cf. A108459, A229233, A229261, A282190, A308490.
Cf. A350722, A351180.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)
  • PARI
    a(n) = sum(k=0, n, k^(k+n)*stirling(n, k, 2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k/k!)))

Formula

E.g.f.: Sum_{k>=0} (k * (exp(k*x) - 1))^k / k!.
a(n) ~ c * r^(2*n) * (1 + exp(1 + 1/r))^n * n^(2*n) / exp(2*n), where r = 0.942405403803582963024019065398882138211529545249588032669864757847... is the root of the equation r*(1 + exp(-1 - 1/r)) * LambertW(-exp(-1/r)/r) = -1 and c = 0.94346979328254581112250921799629823027437848684764713214690470878402... - Vaclav Kotesovec, Feb 18 2022

A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - n*k*x)).

Original entry on oeis.org

1, 1, 6, 105, 3568, 204745, 18028266, 2278860535, 394667414016, 90302033890953, 26525942216131330, 9775058594870836861, 4433256936788979640848, 2434899483389881601250937, 1597444746833206096334387802, 1237091666097626095124512681755, 1119205949224015886848972396596736
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 105*x^3/3! + 3568*x^4/4! + 204745*x^5/5! +...
where
A(x) = 1 + x/(1-x) + 2^2*x^2/(2!*(1-2*1*x)*(1-2*2*x)) + 3^3*x^3/(3!*(1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4^4*x^4/(4!*(1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...

Links

Crossrefs

Cf. A119392, A229257, A229258, A229259, A229260, A229261.
Cf. A229233, A229234, A220181, A122399.

Programs

  • Mathematica
    Flatten[{1,Table[n! * Sum[k^n * StirlingS2[n,k] / k!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)
  • PARI
    {a(n)=n!*polcoeff(sum(m=0, 20, m^m*x^m/m!/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=n!^2*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!^2), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=n!*sum(k=0, n, k^n*stirling(n, k, 2)/k!)}
for(n=0, 20, print1(a(n), ", "))

Formula

Sum_{n>=0} a(n)*x^n / n!^2 = Sum_{n>=0} (exp(n*x) - 1)^n / n!^2.
a(n) = n! * Sum_{k=0..n} k^n * Stirling2(n,k) / k!.

A355375 a(n) = Sum_{k=0..n} (-k)^(n-k) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 0, -4, 10, 67, -969, 3341, 86976, -1988704, 14144108, 405611857, -17544321563, 287677263837, 3595470378748, -421298868094940, 14476946230894114, -112253861285434961, -18711849695261432065, 1354595712379990848137, -44436925726445545236496
Offset: 0

Views

Author

Seiichi Manyama, Jun 30 2022

Keywords

Crossrefs

Cf. A229233, A232549, A318183, A355376.

Programs

  • Mathematica
    a[n_] := Sum[(-k)^(n - k) * StirlingS2[n, k], {k, 0, n}]; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Jun 30 2022 *)
  • PARI
    a(n) = sum(k=0, n, (-k)^(n-k)*stirling(n, k, 2));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (1-exp(-k*x))^k/(k^k*k!))))

Formula

E.g.f.: Sum_{k>=0} (1 - exp(-k * x))^k / (k^k * k!).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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