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 1-6 of 6 results.

A052886 Expansion of e.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).

Original entry on oeis.org

0, 1, 3, 19, 207, 3211, 64383, 1581259, 45948927, 1541641771, 58645296063, 2494091717899, 117258952478847, 6038838138717931, 338082244882740543, 20443414320405268939, 1327850160592214291967, 92200405122521276427691, 6815359767190023358085823, 534337135055010788022858379
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Previous name was: A simple grammar.
From the symmetry present in Vladeta Jovovic's Feb 02 2005 formula, it is easy to see that there are no positive even numbers in this sequence. Question: are there any multiples of 5 after the initial zero? Compare also to the comments in A366884. - Antti Karttunen, Jan 02 2024

Links

Crossrefs

Cf. A000108, A054867, A277120, A293379, A366377, A366884.
Cf. A371316, A371317.
Cf. A371329, A376036, A376037.

Programs

  • Maple
    spec := [S,{C=Set(Z,1 <= card),S=Prod(B,C),B=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    CoefficientList[Series[1/2-1/2*(5-4*E^x)^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
a[n_] := Sum[k! StirlingS2[n, k] CatalanNumber[k - 1], {k, 1, n}];
Array[a, 20, 0] (* Peter Luschny, Apr 30 2020 *)
  • PARI
    N=66; x='x+O('x^N);
concat([0],Vec(serlaplace(serreverse(log(1+x-x^2)))))
\\ Joerg Arndt, May 25 2011
  • PARI
    lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = 1+ sum(k=1, n-1, binomial(n,k)*va[k]*va[n-k]);); concat(0, va);} \\ Michel Marcus, Apr 30 2020
  • PARI
    A000108(n) = binomial(2*n, n)/(n+1);
A052886(n) = sum(k=1,n,k!*stirling(n,k,2)*A000108(k-1)); \\ Antti Karttunen, Jan 02 2024

Formula

E.g.f.: (1/2) - (1/2)*(5 - 4*exp(x))^(1/2).
a(n) = 1 + Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k). - Vladeta Jovovic, Feb 02 2005
a(n) = Sum_{k=1..n} k!*Stirling2(n,k)*C(k-1), where C(k) = Catalan numbers (A000108). - Vladimir Kruchinin, Sep 15 2010
a(n) ~ sqrt(5/2)/2 * n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013
Equals the logarithmic derivative of A293379. - Paul D. Hanna, Oct 22 2017
O.g.f.: Sum_{k>=1} C(k-1)*Product_{r=1..k} r*x/(1-r*x), where C = A000108. - Petros Hadjicostas, Jun 12 2020
a(n) = A366377(A000040(n)) = A366884(A098719(n)). - Antti Karttunen, Jan 02 2024
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (exp(x) - 1) / (1 - A(x)).
E.g.f.: Series_Reversion( log(1 + x * (1 - x)) ). (End)

Extensions

New name using e.g.f. by Vaclav Kotesovec, Sep 30 2013

A376041 E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^3)) / (1 - A(x)).

Original entry on oeis.org

0, 1, 9, 191, 6496, 305164, 18317390, 1339293822, 115492112640, 11476262240520, 1291250885222592, 162271449317302632, 22528350072978189600, 3424249337820235241472, 565573503590604522245136, 100864333223422171393303488, 19317041144591537348567168256
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A376038, A376039, A376040.
Cf. A052851, A371327, A376042.
Cf. A376036.

Programs

  • PARI
    a(n) = sum(k=1, n, (3*n+2*k-2)!/(3*n+k-1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} (3*n+2*k-2)!/(3*n+k-1)! * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( (1 - x)^3 * (1 - exp(-x * (1 - x))) ).

A371329 E.g.f. satisfies A(x) = (exp(x/(1 - A(x))) - 1)/(1 - A(x)).

Original entry on oeis.org

0, 1, 5, 58, 1099, 28966, 978669, 40349478, 1964141687, 110251617526, 7010830858753, 498111156585670, 39106669556183475, 3362091299430435846, 314139422902048625717, 31696638229827506705254, 3434797595698979061279727, 397852853779288923308578966
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2024

Keywords

Links

Crossrefs

Cf. A052892, A371330.
Cf. A052886, A376036, A376037.

Programs

  • PARI
    a(n) = sum(k=1, n, (n+2*k-2)!/(n+k-1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n+2*k-2)!/(n+k-1)! * Stirling2(n,k).
E.g.f.: Series_Reversion( (1 - x) * log(1 + x * (1 - x)) ). - Seiichi Manyama, Sep 08 2024

A376037 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^2) - 1) / (1 - A(x)).

Original entry on oeis.org

0, 1, 7, 115, 3047, 111771, 5244555, 299941195, 20239069807, 1574068019851, 138641219870243, 13640672949173403, 1482772864485867399, 176478769995088245595, 22825571074271407363771, 3187825736999237502879019, 478120273969744650293424095
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A371329, A376036.
Cf. A085527, A371371.

Programs

  • PARI
    a(n) = sum(k=1, n, (2*n+2*k-2)!/(2*n+k-1)!*stirling(n, k, 2));

Formula

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

A376034 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^3) - 1) * (1 - A(x))^2.

Original entry on oeis.org

0, 1, 3, 28, 429, 9136, 249315, 8300692, 326261649, 14786485336, 759129218367, 43543567874764, 2759873588979045, 191549117617410736, 14448371199973057659, 1176874833493589697604, 102951969888432809238585, 9626512744249673928398920
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A274265, A376035, A376036.

Programs

  • PARI
    a(n) = sum(k=1, n, (3*n-k-2)!/(3*n-2*k-1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (3*n-k-2)!/(3*n-2*k-1)! * Stirling2(n,k).
E.g.f.: Series_Reversion( (1 - x)^3 * log(1 + x / (1 - x)^2) ).

A376035 E.g.f. satisfies A(x) = exp(x / (1 - A(x))^3) - 1.

Original entry on oeis.org

0, 1, 7, 118, 3205, 120466, 5790619, 339216046, 23443311049, 1867308836986, 168435092561671, 16971155810393302, 1889194092179682061, 230257485553145337106, 30496977601634473249363, 4361533380688447142658046, 669865656003334085318195089
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2024

Keywords

Links

Crossrefs

Cf. A274265, A376034, A376036.
Cf. A052892, A371371.

Programs

  • Mathematica
    Table[Sum[(3*n+k-2)!/(3*n-1)! * StirlingS2[n,k], {k,1,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 10 2024 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(serreverse((1-x)^3*log(1+x)))))
  • PARI
    a(n) = sum(k=1, n, (3*n+k-2)!/(3*n-1)!*stirling(n, k, 2));

Formula

E.g.f.: Series_Reversion( (1 - x)^3 * log(1+x) ).
a(n) = Sum_{k=1..n} (3*n+k-2)!/(3*n-1)! * Stirling2(n,k).
a(n) ~ 3^(4*n-2) * LambertW(2*exp(1/3)/3)^(3*n-1) * n^(n-1) / (sqrt(1 + LambertW(2*exp(1/3)/3)) * exp(n) * 2^(3*n-1) * (3*LambertW(2*exp(1/3)/3) - 1)^(4*n-1)). - Vaclav Kotesovec, Sep 10 2024
Showing 1-6 of 6 results.

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