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 1-3 of 3 results.

A277610 G.f.: 1 / (1 - Sum_{k>=1} k^k * x^k ).

Original entry on oeis.org

1, 1, 5, 36, 339, 3999, 57388, 977577, 19348425, 436886364, 11084633347, 312102694743, 9653262860564, 325242329821529, 11853828646799153, 464582888781914004, 19481645509391087747, 870252961810204549919, 41253445365917239409916, 2068244310629828065675481, 109336176534540098236055769, 6078206718063279979791668252, 354471031348340363987467541507, 21638266052947649126008431859703, 1379839169160669434086676475756260
Offset: 0

Views

Author

Paul D. Hanna, Oct 23 2016

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 36*x^3 + 339*x^4 + 3999*x^5 + 57388*x^6 + 977577*x^7 + 19348425*x^8 + 436886364*x^9 + 11084633347*x^10 +...
The logarithm of the g.f. begins:
log(A(x)) = x + 9*x^2/2 + 94*x^3/3 + 1181*x^4/4 + 17681*x^5/5 + 310308*x^6/6 + 6276565*x^7/7 + 144052445*x^8/8 + 3701113150*x^9/9 + 105252411369*x^10/10 + 3281812541569*x^11/11 + 111313031195216*x^12/12 + 4079782609460013*x^13/13 + 160665945152295921*x^14/14 + 6765274535733165854*x^15/15 +...
which equals the sum
log(A(x)) = (x + 4*x^2 + 27*x^3 + 256*x^4  + 3125*x^5 +...) +
(x^2 + 8*x^3 + 70*x^4 + 728*x^5 + 9027*x^6 + 132136*x^7 +...)/2 +
(x^3 + 12*x^4 + 129*x^5 + 1480*x^6 + 19002*x^7 +...)/3 +
(x^4 + 16*x^5 + 204*x^6 + 2576*x^7 + 34602*x^8 +...)/4 +
(x^5 + 20*x^6 + 295*x^7 + 4080*x^8 + 57635*x^9 +...)/5 +
(x^6 + 24*x^7 + 402*x^8 + 6056*x^9 + 90165*x^10 +...)/6 +
(x^7 + 28*x^8 + 525*x^9 + 8568*x^10 + 134512*x^11 +...)/7 +
... +
(x + 2^2*x^2 + 3^3*x^3 + 4^4*x^4 + 5^5*x^5 +...+ k^k*x^k +...)^n/n +
...

Links

Crossrefs

Cf. A000312, A088342, A277611.

Programs

  • Maple
    G:= 1/(1-Sum(k^k*x^k,k=1..infinity)):
S:= series(G,x,51):
seq(coeff(S,x,j),j=0..50); # Robert Israel, Nov 06 2016
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      add(j^j*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 22 2017
  • Mathematica
    CoefficientList[Series[1/(1 - Sum[k^k * x^k, {k, 1, 20}]), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 06 2016 *)
  • PARI
    {a(n) = polcoeff( 1/(1 - sum(k=1, n+1, k^k * x^k +x*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} k^k * x^k ]^n / n ).
a(n) ~ n^n * (1 + 2*exp(-1)/n). - Vaclav Kotesovec, Nov 06 2016
"INVERT" transform of A000312. - Alois P. Heinz, Sep 22 2017

A277611 Expansion of 1 / (1 - Sum_{k>=1} k^(k-2) * x^k ).

Original entry on oeis.org

1, 1, 2, 6, 27, 180, 1678, 20388, 305331, 5423511, 111282445, 2587931469, 67239205808, 1929910531883, 60636166356164, 2069775112992573, 76268207153351225, 3017346008698599752, 127561003043924116908, 5738904556162964523209, 273764048456544759900846, 13802374108958236134168506, 733335098861491664742838394, 40953333749038944871704984923, 2398217239830108487402017089693, 146949291558772355319517897103987
Offset: 0

Views

Author

Paul D. Hanna, Oct 23 2016

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 180*x^5 + 1678*x^6 + 20388*x^7 + 305331*x^8 + 5423511*x^9 + 111282445*x^10 + 2587931469*x^11 + 67239205808*x^12 +...
such that A(x) = 1 / (1 - Sum_{k>=1} k^(k-2) * x^k ).
The logarithm of the g.f. begins:
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 83*x^4/4 + 746*x^5/5 + 8817*x^6/6 + 129340*x^7/7 + 2261195*x^8/8 + 45815431*x^9/9 + 1054594428*x^10/10 + 27167908186*x^11/11 + 774186515309*x^12/12 + 24174818590638*x^13/13 + 820795732075686*x^14/14 + 30104104733233598*x^15/15 +...
which equals the sum
log(A(x)) = (x + x^2 + 3*x^3 + 16*x^4 + 125*x^5 + 1296*x^6 +...) +
(x^2 + 2*x^3 + 7*x^4 + 38*x^5 + 291*x^6 + 2938*x^7 +...)/2 +
(x^3 + 3*x^4 + 12*x^5 + 67*x^6 + 507*x^7 + 5001*x^8 +...)/3 +
(x^4 + 4*x^5 + 18*x^6 + 104*x^7 + 783*x^8 + 7572*x^9 +...)/4 +
(x^5 + 5*x^6 + 25*x^7 + 150*x^8 + 1130*x^9 + 10751*x^10 +...)/5 +
(x^6 + 6*x^7 + 33*x^8 + 206*x^9 + 1560*x^10 + 14652*x^11 +...)/6 +
(x^7 + 7*x^8 + 42*x^9 + 273*x^10 + 2086*x^11 + 19404*x^12 +...)/7 +
... +
(x + 2^0*x^2 + 3^1*x^3 + 4^2*x^4 + 5^3*x^5 +...+ k^(k-2)*x^k +...)^n/n +
...

Links

Crossrefs

Cf. A088342, A277610.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - Sum[k^(k-2) * x^k, {k, 1, 20}]), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 06 2016 *)
  • PARI
    {a(n) = polcoeff( 1/(1 - sum(k=1, n+1, k^(k-2) * x^k +x*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) ~ n^(n-2) * (1 + 2*exp(-1)/n). - Vaclav Kotesovec, Nov 06 2016
a(0) = 1; a(n) = Sum_{k=1..n} k^(k-2) * a(n-k). - Ilya Gutkovskiy, Feb 07 2020

A332238 a(n) = n^(n-1) - Sum_{k=1..n-1} k^(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 6, 47, 493, 6446, 101009, 1846631, 38617674, 909844075, 23858239469, 689399172870, 21769608499937, 745964574859679, 27570932237831874, 1093403260892542195, 46315049663202237389, 2087041161850908432022, 99691702658041778953249, 5031814773759672418067623
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 07 2020

Keywords

Crossrefs

Cf. A000169, A003319, A088342, A296715, A332239.

Programs

  • Mathematica
    a[n_] := a[n] = n^(n - 1) - Sum[k^(k - 1) a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]
nmax = 20; CoefficientList[Series[1 - 1/(1 + Sum[k^(k - 1) x^k, {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

Formula

G.f.: 1 - 1 / (1 + Sum_{k>=1} k^(k-1) * x^k).
Showing 1-3 of 3 results.

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