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.

A185391 a(n) = Sum_{k=0..n} A185390(n,k) * k.

Original entry on oeis.org

0, 1, 10, 114, 1556, 25080, 468462, 9971920, 238551336, 6339784320, 185391061010, 5917263922944, 204735466350780, 7633925334590464, 305188474579874550, 13023103577435351040, 590850477768105474128, 28401410966866912051200, 1441935117039649859464986
Offset: 0

Views

Author

Geoffrey Critzer, Feb 09 2012

Keywords

Comments

The total number of elements, x in the domain of definition of all partial functions on n labeled objects such that for all i in {1,2,3,...} (f^i)(x) is defined.

Links

Crossrefs

Cf. A000312, A001865, A076728, A185390.

Programs

  • Mathematica
    nn=20; tx=Sum[n^(n-1) x^n/n!,{n,1,nn}]; txy=Sum[n^(n-1) (x y)^n/n!, {n,1,nn}]; f[list_] := Select[list, #>0&];
  D[Range[0,nn]! CoefficientList[Series[Exp[tx]/(1-txy),{x,0,nn}],x],y]/.y->1
  • PARI
    {a(n) = (n+1)^(n+1)-sum(k=1, n+1, binomial(n+1, k)*k^k*(n+1-k)^(n+1-k))/(n+1)} \\ Seiichi Manyama, Jun 01 2019

Formula

a(n) = (n+1)^(n+1) - A001865(n+1). - Seiichi Manyama, Jun 01 2019

A186390 Adjusted joint rank sequence of (f(i)) and (g(j)) with f(i) after g(j) when f(i)=g(j), where f(i)=6i and g(j)=j(j+1)/2 (triangular number). Complement of A186389.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 16, 19, 21, 24, 28, 31, 34, 38, 42, 46, 50, 54, 59, 64, 68, 73, 79, 84, 89, 95, 101, 107, 113, 119, 126, 133, 139, 146, 154, 161, 168, 176, 184, 192, 200, 208, 217, 226, 234, 243, 253, 262, 271, 281, 291, 301, 311, 321, 332, 343, 353, 364, 376, 387, 398, 410, 422, 434, 446, 458, 471, 484, 496, 509, 523, 536, 549, 563, 577, 591, 605, 619, 634, 649, 663, 678, 694, 709
Offset: 1

Views

Author

Clark Kimberling, Feb 19 2011

Keywords

Examples

			First, write
......6.....12..18....24..30. (6i)
1..3..6..10...15...21...28... (triangular)
Then replace each number by its rank, where ties are settled by ranking 6i after the triangular:
a=(4,6,8,10,12,14,15,17,...)=A186389
b=(1,2,3,5,7,9,11,13,16,...)=A186390.

Crossrefs

Cf. A186387, A186388, A185390.

Programs

A066320 Triangle read by rows: T(n, k) = binomial(n, k)*k^k*(n-k)^(n-k-1) k=0..n-1.

Original entry on oeis.org

1, 2, 2, 9, 6, 12, 64, 36, 48, 108, 625, 320, 360, 540, 1280, 7776, 3750, 3840, 4860, 7680, 18750, 117649, 54432, 52500, 60480, 80640, 131250, 326592, 2097152, 941192, 870912, 945000, 1146880, 1575000, 2612736, 6588344, 43046721
Offset: 1

Views

Author

Christian G. Bower, Dec 13 2001

Keywords

Examples

			Triangle starts:
  [1][      1]
  [2][      2,      2]
  [3][      9,      6,     12]
  [4][     64,     36,     48,    108]
  [5][    625,    320,    360,    540,    1280]
  [6][   7776,   3750,   3840,   4860,    7680,   18750]
  [7][ 117649,  54432,  52500,  60480,   80640,  131250,  326592]
  [8][2097152, 941192, 870912, 945000, 1146880, 1575000, 2612736, 6588344]

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 68 (2.1.43).

Crossrefs

T = n * A185390 after proper alignment of offsets.
Columns 1, 2: A000169, A055541.
Main diagonal: A055897.
Row sums give A000312.

Programs

  • Julia
    # Assuming offset (n=1, k=1).
T(n, k) = binomial(n-1, k-1)*(k-1)^(k-1)*n*(n-k+1)^(n-k-1)
for n in 1:9 (println([n], [T(n, k) for k in 1:n])) end
# Peter Luschny, Jan 12 2024

Formula

E.g.f.: -LambertW(-y)/(1+LambertW(-x*y)). - Vladeta Jovovic, Jan 26 2006
T(n, k) = n*binomial(n-1, k-1)*(k-1)^(k-1)*(n-k+1)^(n-k-1) assuming offset (1, 1). - Peter Luschny, Jan 12 2024
Showing 1-3 of 3 results.

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