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.

Previous Showing 11-20 of 20 results.

A384343 Expansion of Product_{k>=1} (1 + k*x)^((1/2)^(k+1)).

Original entry on oeis.org

1, 1, -1, 3, -14, 86, -650, 5822, -60287, 708873, -9334633, 136142011, -2179136696, 37987580268, -716513806824, 14540745561432, -315936103907094, 7318039354370826, -180020739049731594, 4687207255550122014, -128782014195949550724, 3723598212075752653284, -113023054997369519314572
Offset: 0

Views

Author

Seiichi Manyama, May 26 2025

Keywords

Crossrefs

Cf. A381890, A384344, A384345.
Cf. A000670, A084784.

Programs

  • Mathematica
    terms = 25; A[] = 1; Do[A[x] = -A[x] + 2*((1 + x)*A[x/(1 + x)])^(1/2) + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Vaclav Kotesovec, May 29 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, j!*stirling(k, j, 2))*x^k/k)))

Formula

G.f. A(x) satisfies A(x) = (1+x)^(1/2) * A(x/(1+x))^(1/2).
G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A000670(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A084784.
a(n) ~ (-1)^(n+1) * (n-1)! / (2*log(2)^(n+1)). - Vaclav Kotesovec, May 29 2025

A088791 Coefficient of x^n in A(x)^n is A000670(n), which gives preferential arrangements of n labeled elements.

Original entry on oeis.org

1, 1, 1, 2, 8, 46, 337, 2976, 30627, 359222, 4725968, 68903766, 1102712316, 19219507328, 362428546833, 7352854216056, 159705991698432, 3697928742242694, 90933523698184947, 2366758931071064064
Offset: 0

Views

Author

Paul D. Hanna, Oct 16 2003

Keywords

Links

Crossrefs

Cf. A000670, A084784.

Programs

  • Mathematica
    nmax = 19; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^2 - (A[x A[x]] + x A[x]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)
  • PARI
    {a(n)=local(A, m); if(n<1, n==0, m=1; A=1+x; for(i=1, n, A=(subst(A, x, x*A+x*O(x^n)) + x*A)/A); polcoeff(A, n))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Feb 11 2015

Formula

G.f. satisfies: A(x)^2 = A(x*A(x)) + x*A(x).
a(n) ~ (n-1)! / (4 * (log(2))^(n+1)). - Vaclav Kotesovec, Feb 12 2015
O.g.f.: A(x) = x/( series reversion x*B(x) ), where B(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 137*x^5 + ... is the o.g.f. of A084784. - Peter Bala, Jun 23 2015

A195983 G.f.: Product_{n>=1} [Product_{k=0..2*n} (1 - k*x)^(-(-1)^k*binomial(2*n,k))]^(1/n).

Original entry on oeis.org

1, 0, 1, 2, 7, 32, 189, 1382, 12090, 123220, 1433760, 18750856, 272243150, 4344694984, 75590125814, 1423923579636, 28872306351182, 626986388773656, 14518030696953650, 357068944503961140, 9296168815245928682, 255409012110904594768, 7385107814017356745382
Offset: 0

Views

Author

Paul D. Hanna, Sep 25 2011

Keywords

Comments

Compare g.f. to the product:
exp(x) = Product{n>=1} [Product_{k=0..n} (1 + k*x)^(-(-1)^k*binomial(n,k))]^(1/n),
more explicitly:
_ exp(x) = [(1+x)^1/(1+0*x)^1] * [(1+x)^2/((1+0*x)^1*(1+2*x)^1)]^(1/2) * [(1+x)^3*(1+3*x)^1/((1+0*x)^1*(1+2*x)^3)]^(1/3) * [(1+x)^4*(1+3*x)^4/((1+0*x)^1*(1+2*x)^6*(1+4*x)^1)]^(1/4) * ...

Examples

			G.f.: A(x) = 1 + x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 189*x^6 + 1382*x^7 +...
where
A(x) = [(1-x)^2/((1-0*x)^1*(1-2*x)^1)] * [(1-x)^4*(1-3*x)^4/((1-0*x)^1*(1-2*x)^6*(1-4*x)^1)]^(1/2) * [(1-x)^6*(1-3*x)^20*(1-5*x)^6/((1-0*x)^1*(1-2*x)^15*(1-4*x)^15*(1-6*x)^1)]^(1/3) * ...
Also,
log(A(x)) = 2*x^2/2 + 6*x^3/3 + 26*x^4/4 + 150*x^5/5 + 1082*x^6/6 + 9366*x^7/7 +...+ 2*A000670(n)*x^(n+1)/(n+1) +...
thus, the logarithmic derivative equals the series:
A'(x)/A(x) = 2*x/(1-x) + 2*2!*x^2/((1-x)*(1-2*x)) + 2*3!*x^3/((1-x)*(1-2*x)*(1-3*x)) + 2*4!*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) +...

Links

Crossrefs

Cf. A000670, A084784.

Programs

  • PARI
    {a(n)=polcoeff(prod(m=1,n,prod(k=1,2*m,(1-k*x+x*O(x^n))^(-(-1)^k*binomial(2*m,k)/m))),n)}
  • PARI
    /* Using o.g.f. exp( Sum_{n>=1} 2*A000670(n)*x^(n+1)/(n+1) ): */
{a(n)=polcoeff(exp(intformal(2*sum(m=1, n+1, m!*x^m/prod(k=1, m, 1-k*x+x*O(x^n))))), n)}

Formula

G.f.: A(x) = exp( Sum_{n>=1} 2*A000670(n)*x^(n+1)/(n+1) ) where Sum_{n>=0} A000670(n)*x^n = Sum_{n>=0} n!*x^n/Product_{k=0..n} (1-k*x).
a(n) ~ n! / (n^2 * (log(2))^n). - Vaclav Kotesovec, Nov 19 2014

A384351 Expansion of Product_{k>=1} 1/(1 - k*(k+1)/2 * x)^((1/2)^(k+2)).

Original entry on oeis.org

1, 1, 7, 143, 6140, 455828, 51947988, 8414718996, 1836791273514, 519582028795210, 184852108308617398, 80776494267416227078, 42529172631705836804876, 26553065315757661351020284, 19397441882229095276127402500, 16390942374821715002096327774628
Offset: 0

Views

Author

Seiichi Manyama, May 27 2025

Keywords

Crossrefs

Cf. A084784, A384352, A384353.
Cf. A055203, A262809.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[Sum[Sum[Sum[(-1)^j*Binomial[i, j]*((i - j)*(i - j - 1)/2)^k, {j, 0, i}], {i, 0, 2 k}]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 29 2025 *)
  • PARI
    a262809(n, k) = sum(i=0, k*n, sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a262809(2, k)*x^k/k)))

Formula

G.f.: exp(Sum_{k>=1} A055203(k) * x^k/k).
a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(2*n - 1/2) / (exp(2*n) * log(2)^(2*n+1)). - Vaclav Kotesovec, May 29 2025

A384352 Expansion of Product_{k>=1} 1/(1 - k*(k+1)*(k+2)/6 * x)^((1/2)^(k+3)).

Original entry on oeis.org

1, 1, 32, 5392, 2676188, 2930633692, 5993325199448, 20540879727692152, 109337218761743017718, 854254522610491562826582, 9378640254148405369808277352, 139752461092050444767050922501096, 2747716352285121538660626991038190636, 69628008338488529846443753577404293410060
Offset: 0

Views

Author

Seiichi Manyama, May 27 2025

Keywords

Crossrefs

Cf. A084784, A384351, A384353.
Cf. A062208, A262809.

Programs

  • PARI
    a262809(n, k) = sum(i=0, k*n, sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a262809(3, k)*x^k/k)))

Formula

G.f.: exp(Sum_{k>=1} A062208(k) * x^k/k).

A384353 Expansion of Product_{k>=1} 1/(1 - k*(k+1)*(k+2)*(k+3)/24 * x)^((1/2)^(k+4)).

Original entry on oeis.org

1, 1, 161, 233201, 1388333781, 23407417517205, 900363695229160325, 68584682130559722233525, 9362104205577409136806214275, 2125938144923623062958782871506275, 758178276483321320080629434392636915075, 405630344408921348237973282862682052175313075
Offset: 0

Views

Author

Seiichi Manyama, May 27 2025

Keywords

Crossrefs

Cf. A084784, A384351, A384352.
Cf. A062205, A262809.

Programs

  • PARI
    a262809(n, k) = sum(i=0, k*n, sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a262809(4, k)*x^k/k)))

Formula

G.f.: exp(Sum_{k>=1} A062205(k) * x^k/k).

A243435 O.g.f.: exp( Integral Sum_{n>=1} n! * n^(n-1) * x^(n-1) / Product_{k=1..n} (1 - k*n*x) dx ).

Original entry on oeis.org

1, 1, 3, 29, 686, 30552, 2191262, 230356646, 33349943718, 6359939775042, 1545000640114242, 465750550069828422, 170603300462464687996, 74630981535308266499848, 38429419191031108995080412, 23008323194727484508595195772, 15848730592891024979096686043722, 12445298391963001703710163766096546
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 29*x^3 + 686*x^4 + 30552*x^5 + 2191262*x^6 +...
The logarithmic derivative equals the series:
A'(x)/A(x) = 1/(1-x) + 2!*2*x/((1-2*x)*(1-4*x)) + 3!*3^2*x^2/((1-3*x)*(1-6*x)*(1-9*x)) + 4!*4^3*x^3/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 5!*5^4*x^4/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) +...
Explicitly, the logarithm of the o.g.f. begins:
log(A(x)) = x + 5*x^2/2 + 79*x^3/3 + 2621*x^4/4 + 149071*x^5/5 + 12954365*x^6/6 + 1596620719*x^7/7 + 264914218301*x^8/8 +...

Crossrefs

Cf. A084784, A243440.

Programs

  • PARI
    {a(n)=polcoeff(exp(intformal(sum(m=1, n+1, m!*m^(m-1)*x^(m-1)/prod(k=1, m, 1-m*k*x+x*O(x^n))))), n)}
for(n=0, 30, print1(a(n), ", "))

A243440 O.g.f.: exp( Integral Sum_{n>=1} n! * n^(n-1) * x^(n-1) / Product_{k=1..n} (1 - k*x) dx ).

Original entry on oeis.org

1, 1, 3, 25, 499, 18897, 1158175, 104287909, 12948389505, 2119204222647, 442024984454145, 114447363118335099, 36014003359662761889, 13536516384259740525435, 5989775500211255393302197, 3082008257212085146469317911, 1824650971940959528920159955650, 1231558332755627626667173051846452
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 25*x^3 + 499*x^4 + 18897*x^5 + 1158175*x^6 +...
The logarithmic derivative equals the series:
A'(x)/A(x) = 1/(1-x) + 2!*2*x/((1-x)*(1-2*x)) + 3!*3^2*x^2/((1-x)*(1-2*x)*(1-3*x)) + 4!*4^3*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + 5!*5^4*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) +...
Explicitly, the logarithm of the o.g.f. begins:
log(A(x)) = x + 5*x^2/2 + 67*x^3/3 + 1889*x^4/4 + 91771*x^5/5 + 6828545*x^6/6 + 721578187*x^7/7 + 102730470449*x^8/8 +...

Crossrefs

Cf. A084784, A243435.

Programs

  • PARI
    {a(n)=polcoeff(exp(intformal(sum(m=1, n+1, m!*m^(m-1)*x^(m-1)/prod(k=1, m, 1-k*x+x*O(x^n))))), n)}
for(n=0, 30, print1(a(n), ", "))

A384408 Expansion of Product_{k>=1} 1/(1 - k^3 * x)^((1/2)^(k+1)).

Original entry on oeis.org

1, 13, 2426, 2393226, 7056543721, 46153703519501, 564874416706639304, 11596724623199364432312, 369937054535706501459633546, 17326810763609633232550088712162, 1140582994940898154002780391375267884, 101920298764725526200442366857326292990348
Offset: 0

Views

Author

Seiichi Manyama, May 28 2025

Keywords

Crossrefs

Cf. A084784, A384410.
Cf. A249941.

Programs

  • PARI
    a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a000670(3*k)*x^k/k)))

Formula

G.f.: exp(Sum_{k>=1} A000670(3*k) * x^k/k).
a(n) ~ sqrt(Pi) * 3^(3*n + 1/2) * n^(3*n - 1/2) / (sqrt(2) * exp(3*n) * log(2)^(3*n+1)). - Vaclav Kotesovec, May 29 2025

A384410 Expansion of Product_{k>=1} 1/(1 - k^5 * x)^((1/2)^(k+1)).

Original entry on oeis.org

1, 541, 51270122, 76788748015146, 669464791102102157065, 21339839181227035325658510557, 1900606380926543510490023912037413624, 396633271551441702901523258702004560154006264, 171270169295129060094464591065561066259566766138488074
Offset: 0

Views

Author

Seiichi Manyama, May 28 2025

Keywords

Crossrefs

Cf. A084784, A384408.
Cf. A000670.

Programs

  • PARI
    a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
my(N=10, x='x+O('x^N)); Vec(exp(sum(k=1, N, a000670(5*k)*x^k/k)))

Formula

G.f.: exp(Sum_{k>=1} A000670(5*k) * x^k/k).
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.