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 16 results. Next

A121452 Exponential generating function (1-x^2)^(-1/x).

Original entry on oeis.org

1, 1, 1, 4, 13, 71, 391, 2836, 21729, 198829, 1939501, 21515836, 254169301, 3319328299, 45979476635, 691443303916, 10979537304961, 186915474027321, 3345563762493049, 63613875064443796, 1266776073045809341
Offset: 0

Views

Author

Vladeta Jovovic, Sep 07 2006

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 71*x^5/5! + 391*x^6/6! + ... such that 1/A(x)^x = 1 - x^2.
The logarithm of the e.g.f. is given by the series:
log(A(x)) = (1-x)*(x + x*(x+x/2) + x^2*(x+x^2/2+x^3/3) + x^3*(x+x^2/2+x^3/3+x^4/4)  + x^4*(x+x^2/2+x^3/3+x^4/4+x^5/5) + ...)
log(A(x)) = x + x^3/2 + x^5/3 + x^7/4 + x^9/5 + ...

Links

Crossrefs

Cf. A087761.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(1-x^2)^(-1/x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Sep 28 2013 *)
  • PARI
    {a(n)=n!*polcoeff((1-x^2 +x^2*O(x^n))^(-1/x),n)}
  • PARI
    {a(n)=n!*polcoeff(exp((1-x)*sum(m=0,n,x^m*sum(k=1,m+1,x^k/k)+x*O(x^n))),n)} /* Paul D. Hanna, May 03 2012 */
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^2)/x))) \\ Seiichi Manyama, May 01 2022
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, (i+1)\2, (2*j-1)/j*v[i-2*j+2]/(i-2*j+1)!)); v; \\ Seiichi Manyama, May 01 2022
  • PARI
    a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, n-2*k, 1))/(n-k)!); \\ Seiichi Manyama, May 01 2022

Formula

E.g.f.: (1-x^2)^(-1/x) = Sum_{n>=0} a(n)*x^n/n!.
E.g.f.: exp( (1-x) * Sum_{n>=0} x^n * Sum_{k=1..n+1} x^k/k ). - Paul D. Hanna, May 03 2012
a(n) = n!*Sum_{m=floor((n+1)/2)..n} (-1)^(n-m)*Stirling1(m,2*m-n)/m!. - Vladimir Kruchinin, Mar 09 2013
a(n) ~ n! / 2. - Vaclav Kotesovec, Feb 25 2014
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor((n+1)/2)} (2*k-1)/k * a(n-2*k+1)/(n-2*k+1)!. - Seiichi Manyama, May 01 2022

Extensions

More terms from Klaus Brockhaus, Sep 10 2006

A305306 Expansion of e.g.f. 1/(1 + log(1 - x)/(1 - x)).

Original entry on oeis.org

1, 1, 5, 35, 324, 3744, 51902, 839362, 15513096, 322550616, 7451677632, 189366303840, 5249764639248, 157666361452560, 5099445234111888, 176713626295062384, 6531995374500741888, 256537368987293878272, 10667901271715707803264, 468261481657502075856768, 21635865693957558515860224
Offset: 0

Views

Author

Ilya Gutkovskiy, May 29 2018

Keywords

Comments

a(n)/n! is the invert transform of [1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ...].

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 324*x^4/4! + 3744*x^5/5! + 51902*x^6/6! + ...

Links

Crossrefs

Cf. A000254, A001008, A002805, A007840, A128044, A128045, A305307.
Cf. A006153, A087761.

Programs

  • Maple
    H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
b:= proc(n) option remember; `if`(n=0, 1,
      add(H(j)*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018
  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[1/(1 - Sum[HarmonicNumber[k] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[HarmonicNumber[k] a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 20}]
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)/(1-x)))) \\ Seiichi Manyama, May 10 2023

Formula

E.g.f.: 1/(1 - Sum_{k>=1} (A001008(k)/A002805(k))*x^k).
a(n) ~ n! / ((1/LambertW(1)^2 - 1) * (1 - LambertW(1))^n). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A006153(k). - Seiichi Manyama, May 10 2023

A235385 E.g.f.: exp( Sum_{n>=1} H(n) * x^(2*n-1)/(2*n-1) ) where H(n) is the n-th harmonic number.

Original entry on oeis.org

1, 1, 1, 4, 13, 75, 415, 3160, 24545, 233509, 2323165, 26599780, 321545365, 4312503655, 61219938915, 942271981240, 15340303899265, 266671144108265, 4892612440317145, 94840103781865060, 1934826541931748925, 41387703314570495875, 928953515444722956775, 21738929496091877729400
Offset: 0

Views

Author

Paul D. Hanna, Jan 08 2014

Keywords

Comments

Compare to: exp( Sum_{n>=1} x^(2*n-1)/(2*n-1) ) = sqrt(1-x^2)/(1-x).

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 75*x^5/5! +...
where
log(A(x)) = x + (1+1/2)*x^3/3 + (1+1/2+1/3)*x^5/5 + (1+1/2+1/3+1/4)*x^7/7 + (1+1/2+1/3+1/4+1/5)*x^9/9 + (1+1/2+1/3+1/4+1/5+1/6)*x^11/11 +...
Explicitly,
log(A(x)) = x + 1/2*x^3 + 11/30*x^5 + 25/84*x^7 + 137/540*x^9 + 49/220*x^11 + 363/1820*x^13 + 761/4200*x^15 +...
Equivalently,
log(A(x)) = x + 3*x^3/3! + 44*x^5/5! + 1500*x^7/7! + 92064*x^9/9! + 8890560*x^11/11! + 1241982720*x^13/13! + 236938141440*x^15/15! +...

Crossrefs

Cf. A235685, A087761.

Programs

  • PARI
    {H(n)=sum(k=1,n,1/k)}
{a(n)=local(A=1);A=exp(sum(k=1,n\2+1,H(k)*x^(2*k-1)/(2*k-1))+x*O(x^n));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A073478 Expansion of (1+x)^(1/(1-x)).

Original entry on oeis.org

1, 1, 2, 9, 44, 290, 2154, 19026, 186752, 2070792, 25119720, 334960560, 4824346152, 75100568088, 1250180063664, 22235660291880, 419595248663040, 8388866239417920, 176823515257447104, 3923498370610292544
Offset: 0

Views

Author

Vladeta Jovovic, Aug 26 2002

Keywords

Examples

			E.g.f.: (1+x)^(1/(1-x)) = 1 + x + 2*x^2/2! + 9*x^3/3! + 44*x^4/4! + 290*x^5/5! + 2154*x^6/6! + 19026*x^7/7! + 186752*x^8/8! + 2070792*x^9/9! + ...
which may be written as
(1+x)^(1/(1-x)) = exp(x + x^2*(1+x)/2 + x^3*(1+x+x^2)/3 + x^4*(1+x+x^2+x^3)/4 + x^5*(1+x+x^2+x^3+x^4)/5 + ... + x^n*((1-x^n)/(1-x))/n + ...).

Links

Crossrefs

Cf. A007113, A007120, A087761.

Programs

  • Mathematica
    CoefficientList[Series[(1+x)^(1/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 21 2014 *)
  • PARI
    {a(n)=n!*polcoeff((1+x +x*O(x^n))^(1/(1-x)),n)} \\ Paul D. Hanna, Jan 08 2014
  • PARI
    {a(n)=local(A);A=exp(sum(m=1,n,sum(k=1,m,-(-1)^k/k)*x^m)+x*O(x^n)); n!*polcoeff(A,n)} \\ Paul D. Hanna, Jan 08 2014

Formula

E.g.f.: exp( Sum_{n>=1} x^n * Sum_{k=1..n} -(-1)^k/k ). - Paul D. Hanna, Jan 08 2014
E.g.f.: exp( Sum_{n>=1} x^n * ((1-x^n)/(1-x)) / n ). - Paul D. Hanna, Nov 24 2024
a(n) ~ (log(2))^(1/4) * exp(2*sqrt(n*log(2)) - n - 1/2) * n^(n-1/4). - Vaclav Kotesovec, Apr 21 2014

Extensions

More terms from Robert G. Wilson v, Aug 28 2002

A300491 Expansion of e.g.f. log(1 - log(1 - x)/(1 - x)).

Original entry on oeis.org

0, 1, 2, 4, 9, 28, 140, 936, 6902, 54160, 467784, 4578000, 50434032, 609309504, 7921524624, 110242136928, 1643101763760, 26192405980416, 444523225673472, 7989603260143104, 151483589818925184, 3022296286833907200, 63326051483436129024, 1390571693776506751488
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 07 2018

Keywords

Comments

Logarithmic transform of A000254.

Examples

			log(1 - log(1 - x)/(1 - x)) = x/1! + 2*x^2/2! + 4*x^3/3! + 9*x^4/4! + 28*x^5/5! + 140*x^6/6! + 936*x^7/7! + ...

Links

Crossrefs

Cf. A000254, A008405, A087761.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<2, n, n*b(n-1)+(n-1)!) end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-add(
      a(j)*j*binomial(n, j)*b(n-j), j=1..n-1)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 07 2018
  • Mathematica
    nmax = 23; CoefficientList[Series[Log[1 - Log[1 - x]/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n! HarmonicNumber[n] - Sum[k Binomial[n, k] (n - k)! HarmonicNumber[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]

A307419 Triangle of harmonic numbers T(n, k) = [t^n] Gamma(n+k+t)/Gamma(k+t) for n >= 0 and 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 11, 9, 1, 0, 50, 71, 18, 1, 0, 274, 580, 245, 30, 1, 0, 1764, 5104, 3135, 625, 45, 1, 0, 13068, 48860, 40369, 11515, 1330, 63, 1, 0, 109584, 509004, 537628, 203889, 33320, 2506, 84, 1, 0, 1026576, 5753736, 7494416, 3602088, 775929, 81900, 4326, 108, 1
Offset: 0

Views

Author

Vladimir Kruchinin, Apr 08 2019

Keywords

Examples

			Triangle starts:
0: [1]
1: [0,       1]
2: [0,       3,       1]
3: [0,      11,       9,       1]
4: [0,      50,      71,      18,       1]
5: [0,     274,     580,     245,      30,      1]
6: [0,    1764,    5104,    3135,     625,     45,     1]
7: [0,   13068,   48860,   40369,   11515,   1330,    63,    1]
8: [0,  109584,  509004,  537628,  203889,  33320,  2506,   84,   1]
9: [0, 1026576, 5753736, 7494416, 3602088, 775929, 81900, 4326, 108, 1]
Col:   A000254, A001706, A001713, A001719, ...

Crossrefs

Row sums are A087761.
Columns are A000007, A000254, A001706, A001713, A001719.
Cf. A001008/A002805.

Programs

  • Maple
    # Note that for n > 16 Maple fails (at least in some versions) to compute the
# terms properly. Inserting 'simplify' or numerical evaluation might help.
A307419Row := proc(n) local ogf, ser; ogf := (n, k) -> GAMMA(n+k+x)/GAMMA(k+x);
ser := (n, k) -> series(ogf(n,k),x,k+2); seq(coeff(ser(n,k),x,k),k=0..n) end: seq(A307419Row(n), n=0..9);
# Alternatively by the egf for column k:
A307419Col := proc(n, len) local f, egf, ser; f := (n,x) -> (log(1-x)/(x-1))^n/n!;
egf := (n,x) -> diff(f(n, x), [x$n]); ser := n -> series(egf(n, x), x, len);
seq(k!*coeff(ser(n), x, k), k=0..len-1) end:
seq(print(A307419Col(k, 10)), k=0..9); # Peter Luschny, Apr 12 2019
T := (n, k) -> add((-1)^(n-j)*binomial(j, k)*Stirling1(n, j)*k^(j-k), j = k..n):
seq(seq(T(n,k), k = 0..n), n = 0..9); # Peter Luschny, Jun 09 2022
  • Mathematica
    f[n_, x_] := f[n, x] = D[(Log[1 - x]/(x - 1))^n/n!, {x, n}];
T[n_, k_] := (n - k)! SeriesCoefficient[f[k, x], {x, 0, n - k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 13 2019 *)
  • Maxima
    T(n,k):=n!*sum((binomial(k+i-1,i)*abs(stirling1(n-i,k)))/(n-i)!,i,0,n-k);
  • Maxima
    taylor((1-t)^(-x/(1-t)),t,0,7,x,0,7);
  • Maxima
    T(n,k):=coeff(taylor(gamma(n+k+t)/gamma(k+t),t,0,10),t,k);
  • PARI
    T(n, k) = n!*sum(i=0, n-k, abs(stirling(n-i, k, 1))*binomial(i+k-1, i)/(n-i)!); \\ Michel Marcus, Apr 13 2019

Formula

E.g.f.: A(t,x) = (1-t)^(-x/(1-t)).
T(n, k) = n!*Sum_{L1+L2+...+Lk=n} H(L1)H(L2)...H(Lk) with Li > 0, where H(n) are the harmonic numbers A001008.
T(n, k) = n!*Sum_{i=0..n-k} abs(Stirling1(n-i, k))/(n-i)!*binomial(i+k-1, i).
T(n, k) = k! [x^k] (d^n/dx^n) ((log(1-x)/(x-1))^n/n!), the e.g.f. for column k where Col(k) = [T(n+k, k) for n = 0, 1, 2, ...]. - Peter Luschny, Apr 12 2019
T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(j, k)*Stirling1(n, j)*k^(j-k). - Peter Luschny, Jun 09 2022

A308346 Expansion of e.g.f. 1/(1 - x)^log(1 - x).

Original entry on oeis.org

1, 0, -2, -6, -10, 20, 352, 2772, 18132, 104400, 469608, 238920, -35811048, -730972944, -11436661728, -164609993520, -2294024595312, -31488879303552, -426338226719904, -5626751283423072, -70000948158061728, -745703905072996800, -4142683990211677440, 110386551348875714880
Offset: 0

Views

Author

Ilya Gutkovskiy, May 21 2019

Keywords

Links

Crossrefs

Cf. A000254, A008405, A009199, A052517, A067994, A086660, A087761.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^Log(1/(1-x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 21 2019
  • Maple
    E:= 1/(1-x)^log(1-x):
S:= series(E,x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, May 22 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - x)^Log[1 - x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HermiteH[k, 0], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = -2 Sum[(k - 1)! HarmonicNumber[k - 1] Binomial[n - 1, k - 1] a[n - k], {k, 2, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
  • PARI
    a(n) = sum(k=0, n, abs(stirling(n, k, 1))*polhermite(k, 0)); \\ Michel Marcus, May 21 2019
  • Sage
    m = 30; T = taylor((1-x)^log(1/(1-x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 21 2019

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A067994(k).

A347339 E.g.f.: exp( (exp(x) - 1) * exp(exp(x) - 1) ).

Original entry on oeis.org

1, 1, 4, 20, 123, 902, 7656, 73509, 785154, 9213324, 117624569, 1621028312, 23959376436, 377730250003, 6322478398476, 111904530008040, 2087093471665987, 40891426070289970, 839329531471890724, 18004595602417946685, 402747680140030433886, 9376084240910510840672, 226760664399026618376569
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 27 2021

Keywords

Comments

Exponential transform of A138378.
Stirling transform of A000248.

Links

Crossrefs

Cf. A000110, A000248, A000258, A005493, A005727, A087761, A138378.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*j*binomial(n-1, j-1), j=1..n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Aug 27 2021
  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[(Exp[x] - 1) Exp[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (BellB[k + 1] - BellB[k]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(exp((exp(x)-1)*exp(exp(x)-1)))) \\ Michel Marcus, Aug 27 2021

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * (Bell(k+1) - Bell(k)) * a(n-k).
a(n) = Sum_{k=0..n} Stirling2(n,k) * A000248(k).

A235776 E.g.f.: exp( Sum_{n>=1} x^(2*n) * Sum_{k=1..n} 1/k^2 ).

Original entry on oeis.org

1, 2, 42, 2000, 170660, 22741992, 4344779208, 1123066676160, 376718037181200, 158895919895100960, 82222168141278271392, 51172838316787466103552, 37687233953299944682503744, 32399590493755848692815785600, 32140659218911596667452247171200
Offset: 0

Views

Author

Paul D. Hanna, Jan 15 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x^2/2! + 42*x^4/4! + 2000*x^6/6! + 170660*x^8/8! +...
such that
log(A(x)) = x^2 + (1+1/4)*x^4 + (1+1/4+1/9)*x^6 + (1+1/4+1/9+1/16)*x^8 + (1+1/4+1/9+1/16+1/25)*x^10 + (1+1/4+1/9+1/16+1/25+1/36)*x^12 +...
Explicitly,
log(A(x)) = x^2 + 5/4*x^4 + 49/36*x^6 + 205/144*x^8 + 5269/3600*x^10 + 5369/3600*x^12 + 266681/176400*x^14 +...+ [Sum_{k=1..n} 1/k^2]*x^(2*n) +...

Links

Crossrefs

Cf. A087761, A235385.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[PolyLog[2,x]/(1-x)], {x, 0, nmax}], x] * (2*Range[0, nmax])! (* Vaclav Kotesovec, Oct 28 2024 *)
  • PARI
    {a(n)=local(A=1); A=exp(sum(m=1, n\2+1, sum(k=1, m, 1/k^2)*x^(2*m))+x*O(x^n)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(2*n), ", "))

A336289 a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * (k-1)! * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, 1, 5, 55, 1054, 31046, 1299386, 73211510, 5338080280, 488727800664, 54865512897432, 7408400404206792, 1184230737883333680, 221121985937352261360, 47683177920267470877648, 11758982455716373002624816, 3287966057434181416523799936
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 16 2020

Keywords

Crossrefs

Cf. A001008, A002805, A074707, A087761, A235385, A235685, A336290.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = n! Sum[Binomial[n - 1, k - 1] (k - 1)! HarmonicNumber[k] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[Exp[Sum[HarmonicNumber[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 16; CoefficientList[Series[Exp[Log[1 - x]^2/2 + PolyLog[2, x]], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} H(n) * x^n / n).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(log(1 - x)^2 / 2 + polylog(2,x)).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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