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.

A093963 Antidiagonal sums of array in A093966.

Original entry on oeis.org

1, 3, 8, 20, 49, 123, 312, 824, 2221, 6235, 17904, 53348, 162545, 511747, 1645776, 5448600, 18404189, 63794611, 225353368, 814801812, 2999022641, 11274044075, 43100574472, 167987074584, 665229445293, 2681607587627, 10973746015456
Offset: 1

Views

Author

Ralf Stephan, Apr 20 2004

Keywords

Links

Crossrefs

Cf. A093964, A093965.

Programs

  • Mathematica
    A[n_, k_]:= A[n, k]= If[n==1, 1, If[k==1, n, If[2<=kG. C. Greubel, Dec 29 2021 *)
  • Sage
    @CachedFunction
def A(n, k):
    if (n==1): return 1
    elif (k==1): return n
    elif (2 <= k < n+1): return factorial(k)*binomial(n, k) + sum( j*factorial(j)*binomial(n, j) for j in (1..k-1) )
    else: return sum( j*factorial(j)*binomial(n, j) for j in (1..n) )
@CachedFunction
def a(n): return sum( A(k, n-k+1) for k in (1..n) )
[a(n) for n in (1..30)] # G. C. Greubel, Dec 29 2021

Formula

Conjecture: 2*a(n) -5*a(n-1) -(n+2)*a(n-2) +2*(n+6)*a(n-3) +(n-13)*a(n-4) -4*(n-3)*a(n-5) +2*(n-3)*a(n-6) = 0. - R. J. Mathar, Nov 10 2013

A093964 a(n) = Sum_{k=1..n} k*k!*C(n,k).

Original entry on oeis.org

0, 1, 6, 33, 196, 1305, 9786, 82201, 767208, 7891281, 88776910, 1085051121, 14322674796, 203121569833, 3080677142466, 49764784609065, 853110593298256, 15469738758475041, 295858753755835158, 5951981987323272001, 125652953065713520020, 2777591594084193600441
Offset: 0

Views

Author

Ralf Stephan, Apr 20 2004

Keywords

Comments

Limit to which the columns of array A093966 converge.
Number of objects in all permutations of n objects taken 1,2,...,n at a time. Example: a(2)=6 because the permutations of {a,b} taken 1 and 2 at a time are: a,b,ab and ba, containing altogether 1+1+2+2=6 objects. a(n)=Sum(k*A008279(n,k),k=1..n). - Emeric Deutsch, Aug 16 2006
The number of sequences -where each member is an element in a set consisting of n elements- such that the last member is a repetition of a former member. Example: Set of possible members: {l,r}. Sequences such that the last member is a repetition of a former member: l,l; r,r; l,r,l; l,r,r; r,l,l; r,l,r. a(n)=Sum(k*A008279(n,k),k=1..n). [From Franz Fritsche (ff(AT)simple-line.de), Feb 22 2009]
The total number of elements in all ascending runs (including runs of length 1) over all permutations of {1,2,...,n}. a(2) = 6 because in the permutations [1,2] and [2,1] there are 4 runs of length 1 and 1 run of length 2. a(n) = Sum_{k>=1} A132159(n,k)*k. - Geoffrey Critzer, Feb 24 2014

Examples

			G.f. = x + 6*x^2 + 33*x^3 + 196*x^4 + 1305*x^5 + 9786*x^6 + 82201*x^7 + ...

Links

Crossrefs

Cf. A000522, A001339, A007526, A008279.
Row n=2 of A210472. - Alois P. Heinz, Jan 23 2013

Programs

  • Magma
    [0] cat [n le 2 select 6^(n-1) else n*((n+1)*Self(n-1) - (n-1)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Dec 29 2021
  • Maple
    seq(add(k*n!/(n-k)!,k=1..n),n=0..20); # Emeric Deutsch, Aug 16 2006
# second Maple program:
a:= proc(n) a(n):=`if`(n<2, n, n*((n+1)/(n-1)*a(n-1)-a(n-2))) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 21 2013
  • Mathematica
    nn=21;Range[0,nn]!CoefficientList[Series[D[Exp[y x]/(1-x)^2,y]/.y->1,{x,0,nn}],x] (* Geoffrey Critzer, Feb 24 2014 *)
  • PARI
    a(n)=sum(k=1,n,k*k!*binomial(n,k))
  • Sage
    [factorial(n)*( x*exp(x)/(1-x)^2 ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 29 2021

Formula

E.g.f.: x*exp(x)/(1-x)^2. - Vladeta Jovovic, Apr 24 2004
a(n) = 1 + (n-1)*floor(e*n!) = 1 + (n-1)*A000522(n) = A000522(n+1) - 2*A000522(n) = A001339(n) - A000522(n). - Henry Bottomley, Dec 22 2008
a(n) = n if n < 2, a(n) = n*((n+1)/(n-1)*a(n-1) - a(n-2)) for n >= 2. - Alois P. Heinz, Jan 21 2013
E.g.f.: x*(1- 12*x/(Q(0)+6*x-3*x^2))/(1-x)^2, where Q(k) = 2*(4*k+1)*(32*k^2+16*k+x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
G.f.: conjecture: T(0)/x - 1/x, where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
a(n) = n*a(n-1) + A007526(n), a(0) = 0. - David M. Cerna, May 12 2014

Extensions

a(0) inserted by Alois P. Heinz, Jan 21 2013

A093965 Number of functions of [n] to [n] that simultaneously avoid the patterns 112 and 221.

Original entry on oeis.org

1, 4, 21, 124, 825, 6186, 51961, 484968, 4988241, 56117710, 685883121, 9053657196, 128397320233, 1947359356866, 31457343457065, 539268744978256, 9778739908939041, 187018400758459158, 3762370179964296001, 79427814910357360020, 1755772750650004800441
Offset: 1

Views

Author

Ralf Stephan, Apr 20 2004

Keywords

Links

Crossrefs

Cf. A093966.

Programs

  • Magma
    [n le 2 select 4^(n-1) else n*((n+1)*Self(n-1) - (n-1)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Dec 29 2021
  • Mathematica
    Rest[CoefficientList[Series[x(E^x-x)/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Nov 20 2012 *)
  • PARI
    my(x='x+O('x^66)); Vec(serlaplace(x*(exp(x)-x)/(1-x)^2)) \\ Joerg Arndt, May 11 2013
  • Sage
    [factorial(n)*( x*(exp(x) -x)/(1-x)^2 ).series(x,n+1).list()[n] for n in (1..30)] # G. C. Greubel, Dec 29 2021

Formula

a(n) = A093966(n, n).
From Vaclav Kotesovec, Nov 20 2012: (Start)
E.g.f.: x*(exp(x) - x)/(1-x)^2.
Recurrence: (n-1)*a(n) = n*(n+1)*a(n-1) - (n-1)*n*a(n-2) for n>2.
a(n) ~ n!*n*(e-1). (End)

Extensions

Name changed by Olivier Gérard, Aug 06 2016
Showing 1-3 of 3 results.

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