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 12 results. Next

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!.

A229233 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n*k*x).

Original entry on oeis.org

1, 1, 2, 8, 48, 387, 4043, 52425, 819346, 15133184, 324769270, 7986143453, 222514878501, 6958782341565, 242274294115558, 9324382604206368, 394282071192289024, 18218582054356563951, 915480348188869318723, 49812603754178905560085, 2923492374797360684715882
Offset: 0

Views

Author

Paul D. Hanna, Sep 17 2013

Keywords

Comments

Compare to an o.g.f. of Bell numbers (A000110): Sum_{n>=0} x^n/Product_{k=1..n} (1-k*x).

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 48*x^4 + 387*x^5 + 4043*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/((1-2*1*x)*(1-2*2*x)) + x^3/((1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 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 + 2*x^2/2! + 8*x^3/3! + 48*x^4/4! + 387*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2/(2!*2^2) + (exp(3*x)-1)^3/(3!*3^3) + (exp(4*x)-1)^4/(4!*4^4) + (exp(5*x)-1)^5/(5!*5^5) +...

Links

Crossrefs

Cf. A229234, A220181, A108459, A122399.
Cf. A229258, A229259, A229260, A229261.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[k^(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*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^m)),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0, n, k^(n-k) * stirling(n, k, 2))}
for(n=0,30,print1(a(n),", "))

Formula

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

A307375 Expansion of Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - k^2*x).

Original entry on oeis.org

1, 1, 3, 17, 151, 1893, 31499, 666169, 17351967, 543441005, 20079329875, 861908850561, 42439075349543, 2371469004695797, 149022897087857691, 10448429535366899273, 811758520658841809839, 69463012765807086749949, 6511800419610377560644707, 665560984365147223546851985
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 06 2019

Keywords

Comments

a(n) is the number of partitions of [2n] such that the largest element of each block is even. a(3) = 17: 123456, 1234|56, 12356|4, 124|356, 1256|34, 12|3456, 12|34|56, 12|356|4, 13456|2, 134|256, 134|2|56, 1356|24, 1356|2|4, 14|2356, 156|234, 14|2|356, 156|2|34. - Alois P. Heinz, Jun 10 2023

Links

Crossrefs

Cf. A000670, A135920, A229234, A363589.
Bisection of A290383 (even part).

Programs

  • Maple
    b:= proc(n, x, y) option remember; `if`(n=0, 1, `if`(n::odd, 0,
      b(n-1, y, x+1))+b(n-1, y, x)*x+b(n-1, y, x)*y)
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..19);  # Alois P. Heinz, Jun 10 2023
  • Mathematica
    nmax = 19; CoefficientList[Series[Sum[j! x^j/Product[(1 - k^2 x), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

Original entry on oeis.org

1, 1, 4, 33, 456, 9445, 272448, 10386817, 503758720, 30202999821, 2189000524800, 188349613075393, 18954958449853440, 2203304642871358741, 292675996808408743936, 44022321302156791898625, 7438113993194856900034560, 1401876939543892434209075581
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2020

Keywords

Links

Crossrefs

Cf. A000670, A063170, A086914, A094420, A122704, A122778, A229234, A255927, A301419, A326323, A326324.

Programs

  • Mathematica
    Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(n - k), {k, 0, n}], {n, 1, 17}]]
Table[SeriesCoefficient[Sum[k! x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[n^(n + 1) PolyLog[-n, 1/(n + 1)]/(n + 1), {n, 1, 17}]]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*k!*n^(n-k)); \\ Michel Marcus, Jan 24 2020

Formula

a(n) = [x^n] Sum_{k>=0} k! * x^k / Product_{j=1..k} (1 - n*j*x).
a(n) = n! * [x^n] n / (1 + n - exp(n*x)) for n > 0.
a(n) = n^(n + 1) * Sum_{k>=1} k^n / (n + 1)^(k + 1) for n > 0.
a(n) ~ n! * n^(n+1) / ((n+1) * log(n+1)^(n+1)). - Vaclav Kotesovec, Jun 06 2022

A350725 a(n) = Sum_{k=0..n} k! * k^(n-k) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 1, -4, -2, 274, -3442, -12552, 2108664, -63083232, 87416112, 112192496976, -7487840132544, 174521224997040, 19793498724358032, -3109195219736188416, 209306170972547346816, 2973238556525799866496, -3013574861684426837113728, 456220653756733889826621696
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2022

Keywords

Crossrefs

Cf. A007840, A229234, A320083, A350721, A350726.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * k^(n-k) * StirlingS1[n, k], {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*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)))

Formula

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

A351281 a(n) = Sum_{k=0..n} k! * k^k * Stirling2(n,k).

Original entry on oeis.org

1, 1, 9, 187, 7173, 440611, 39631509, 4910795107, 802015652853, 166948755155971, 43146953460348309, 13555255072473665827, 5087595330217093070133, 2248298922174973220446531, 1155512971750307157457879509, 683392198848998191062416885347
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2022

Keywords

Crossrefs

Cf. A000670, A122399, A229234, A282190, A350722, A351280.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * k^k * StirlingS2[n, k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Feb 06 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*k^k*stirling(n, k, 2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(x)-1))^k)))

Formula

E.g.f.: Sum_{k>=0} (k * (exp(x) - 1))^k.
a(n) ~ exp(exp(-1)/2) * n! * n^n. - Vaclav Kotesovec, Feb 06 2022
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