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

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.

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.

A187755 E.g.f.: Sum_{n>=0} (1 - exp(-n^2*x))^n.

Original entry on oeis.org

1, 1, 31, 3991, 1340251, 929043391, 1153715889691, 2333670966674671, 7180487882511523051, 31919495229412870788031, 196909477461357591810377851, 1632140626754602443266222263951, 17701927686793740884448652685728651, 245721757633690118910277310669218472671
Offset: 0

Views

Author

Paul D. Hanna, Jan 03 2013

Keywords

Examples

			O.g.f.: F(x) = 1 + x + 31*x^2 + 3991*x^3 + 1340251*x^4 + 929043391*x^5 +...
where
F(x) = 1 + x/(1+x) + 2^4*2!*x^2/((1+2^2*1*x)*(1+2^2*2*x)) + 3^6*3!*x^3/((1+3^2*1*x)*(1+3^2*2*x)*(1+3^2*3*x)) + 4^8*4!*x^4/((1+4^2*1*x)*(1+4^2*2*x)*(1+4^2*3*x)*(1+4^2*4*x)) +...
...
E.g.f.: A(x) = 1 + x + 31*x^2/2! + 3991*x^3/3! + 1340251*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2^2*x))^2 + (1-exp(-3^2*x))^3 + (1-exp(-4^2*x))^4 + (1-exp(-5^2*x))^5 + (1-exp(-6^2*x))^6 +...

Links

Crossrefs

Cf. A220181, A203798, A229260.

Programs

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

Formula

O.g.f.: Sum_{n>=0} n^(2*n) * n! * x^n / Product_{k=1..n} (1 + n^2*k*x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n) * k! * Stirling2(n,k).
a(n) == 1 (mod 10) for n>=0.
a(n) == 31 (mod 60) for n>=2.
a(n) ~ c * d^n * (n!)^3 / n, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.192038502554748256318271067254582378566365276592... . - Vaclav Kotesovec, May 08 2014

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)).

A351133 a(n) = Sum_{k=0..n} k! * k^(2*n) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 31, 3992, 1342294, 932514674, 1161340476698, 2356863300156504, 7278091701243797640, 32477694155566998880608, 201155980661221409458717152, 1674230688936725338278370413264, 18235249164492209082483584810706528
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2022

Keywords

Links

Crossrefs

Cf. A006252, A320083, A351134.
Cf. A229260, A351135, A351136.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * k^(2*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 02 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*k^(2*n)*stirling(n, k, 1));
  • PARI
    first(n)=my(x='x+O('x^(n+1))); Vec(serlaplace(sum(k=0, n, log(1+k^2*x)^k)))

Formula

E.g.f.: Sum_{k>=0} log(1 + k^2*x)^k.
a(n) ~ c * d^n * n^(3*n + 1/2), where d = 0.3417329834649268103028466896966197580428514873775849996969994420891... and c = 2.92355271092039591960355156784704285135358... - Vaclav Kotesovec, Feb 03 2022

A242228 a(n) = Sum_{k=1..n} k^(2*n-1) * k! * Stirling2(n,k).

Original entry on oeis.org

1, 17, 1651, 473741, 300257371, 355743405917, 706872713310331, 2182548723605418941, 9894910566488309801851, 63052832687428562206049117, 545439670961897317869306191611, 6226501736967631584015448186252541, 91619831483112536750163352484302283131
Offset: 1

Views

Author

Vaclav Kotesovec, May 08 2014

Keywords

Links

Crossrefs

Cf. A000629, A220179, A229260, A244585.

Programs

  • Mathematica
    Table[Sum[k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}]

Formula

a(n) ~ c * d^n * (n!)^3 / n^2, 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 = 0.15095210978787998524366903417512193343948127919...
E.g.f.: Sum_{k>=1} (exp(k^2*x) - 1)^k / k. - Seiichi Manyama, Jun 19 2024

A242229 a(n) = Sum_{k=0..n} k^(3*n) * k! * StirlingS2(n,k).

Original entry on oeis.org

1, 1, 129, 121171, 421842405, 3921960731851, 80097035486409669, 3154805675402432477371, 218356776433458097793841045, 24765902586563160053438320367371, 4359137561016969073655241431827801509, 1139916274502953599866121961715757905518171
Offset: 0

Views

Author

Vaclav Kotesovec, May 08 2014

Keywords

Links

Crossrefs

Cf. A203798, A242228
Cf. A000670, A122399, A229260.

Programs

  • Mathematica
    Table[Sum[k^(3*n) * k! * StirlingS2[n,k], {k,0,n}], {n,0,20}]
  • PARI
    a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 2)); \\ Seiichi Manyama, Feb 01 2022
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (exp(k^3*x)-1)^k))) \\ Seiichi Manyama, Feb 01 2022
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k^3*x)^k/prod(j=1, k, 1-k^3*j*x))) \\ Seiichi Manyama, Feb 01 2022

Formula

a(n) ~ c * d^n * (n!)^4 / n^(3/2), where d = 20.5647332000203822461493845960846630764635... = r^4*(1+exp(3/r)), r = 0.97762267432285928683132021521727105447350... is the root of the equation (1+exp(-3/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.0600744446309702764688382302731840300640714536...
E.g.f.: Sum_{k>=0} (exp(k^3*x) - 1)^k. - Seiichi Manyama, Feb 01 2022
G.f.: Sum_{k>=0} k! * (k^3*x)^k/Product_{j=1..k} (1 - k^3*j*x). - Seiichi Manyama, Feb 01 2022

Extensions

a(0)=1 prepended by Seiichi Manyama, Feb 01 2022
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