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.

A275707 Number of partial functions f:{1,2,...,n}->{1,2,...,n} such that every element in the domain of definition of f is mapped to a fixed point or to an element that is undefined by f.

Original entry on oeis.org

1, 2, 8, 38, 216, 1402, 10156, 80838, 698704, 6498674, 64579284, 681642238, 7605025720, 89318058858, 1100376445564, 14176837311158, 190498308591264, 2663482511782114, 38667106019619748, 581765160424218606, 9055862445043643656, 145619330650420134362
Offset: 0

Views

Author

Geoffrey Critzer, Aug 06 2016

Keywords

Examples

			G.f. = 1 + 2*x + 8*x^2 + 38*x^3 + 216*x^4 + 1402*x^5 + 10156*x^6 + ...
a(2) = 8 because there are 9 = A000169(3) partial functions on a set with 2 elements and all of them have the stated property except 1->2,2->1.
		

Crossrefs

Programs

  • Maple
    a:= n-> add(binomial(n, k)*add(binomial(n-k, f)*
            (f+k)^(n-k-f), f=0..n-k), k=0..n):
    seq(a(n), n=0..30);  # Alois P. Heinz, Aug 07 2016
  • Mathematica
    nn = 20; Range[0, nn]! CoefficientList[Series[ Exp[z Exp[z]]^2, {z, 0, nn}], z]
    Table[Sum[BellY[n, k, 2 Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
  • PARI
    x='x+O('x^33); Vec(serlaplace(exp(2*x*exp(x)))) \\ Joerg Arndt, Nov 10 2016
    
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*x)^k/(1-k*x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022
    
  • PARI
    a(n) = sum(k=0, n, 2^k*k^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Jul 04 2022

Formula

E.g.f.: A(x)^2 = exp(2*B(x)) where A(x) is the e.g.f. for A000248 and B(x) is the e.g.f. for A000027.
E.g.f.: exp(2*x*exp(x)). - Joerg Arndt, Nov 10 2016
a(0) = 1; a(n) = Sum_{k=1..n} 2*k*binomial(n-1,k-1)*a(n-k). - Ilya Gutkovskiy, Nov 24 2017
From Seiichi Manyama, Jul 04 2022: (Start)
G.f.: Sum_{k>=0} (2 * x)^k/(1 - k*x)^(k+1).
a(n) = Sum_{k=0..n} 2^k * k^(n-k) * binomial(n,k). (End)
a(n) ~ n^(n + 1/2) * exp(2*r*exp(r) - r/2 - n) / (sqrt(2*(1 + 3*r + r^2)) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n)/2^(3/2)). - Vaclav Kotesovec, Jul 06 2022

A187105 Triangle T(n,k) read by rows: number of height-2-restricted finite functions.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 8, 3, 1, 41, 38, 15, 4, 1, 196, 216, 90, 24, 5, 1, 1057, 1402, 633, 172, 35, 6, 1, 6322, 10156, 5028, 1424, 290, 48, 7, 1, 41393, 80838, 44217, 13204, 2745, 450, 63, 8, 1, 293608, 698704, 424434, 134680, 28900, 4776, 658, 80, 9, 1
Offset: 1

Views

Author

Dennis P. Walsh, Mar 04 2011

Keywords

Comments

Triangle T(n,k) with 1 <= k <= n+1 is the number of functions f:[n+1-k]->[n+1] such that f(f(f(x))) is undefined, that is, either f(x) or f(f(x)) is in {n+2-k,...,n+1}. Such functions are called height-2 restricted functions. Note that the null function, which occurs when k=n+1, vacuously satisfies the conditions for a height-2 restricted function, and hence T(n,n+1)=1. The sequence a(n)=T(n,1) is sequence A000248, the number of forests with n nodes and height at most 1. The height of a function f:D->C, with D a proper subset of finite C, is the maximum h such that (f^h)(x) exists for some x in D. A height restricted function f is acyclic since, if x is in a cycle of f, then (f^z)(x) exists for all positive integers z. [Note that [m] denotes the set of the first m positive integers and that f^m denotes the m-fold self-composition of f so that (f^0)(x)=x, (f^1)(x)=f(x),(f^2)(x)=f(f(x)), etc.]

Examples

			Triangle of initial terms:
     1
     1     1
     3     2     1
    10     8     3     1
    41    38    15     4     1
   196   216    90    24     5     1
  1057  1402   633   172    35     6     1
T(4,3) = 15 since there are 15 functions f:[2]->[5] such that either f(x) or f(f(x)) is in {3,4,5}. Using <f(1),f(2)> to denote these functions we have the following 15 functions: <2,3>, <2,4>, <2,5>, <3,1>, <3,3>, <3,4>, <3,5>, <4,1>, <4,3>, <4,4>, <4,5>, <5,1>, <5,3>, <5,4>, <5,5>.
		

Crossrefs

Programs

  • Maple
    seq(seq(sum(binomial(n+1-k,j)*k^j*j^(n+1-k-j),j=0..(n+1-k)),k=1..n),n=1..15); # triangle's right edge of ones is omitted with this program
  • Mathematica
    t[n_, k_] := If[ k == n + 1, 1, Sum[ Binomial[n + 1 - k, j]*k^j*j^(n + 1 - k - j), {j, 0, n + 1 - k}]]; Table[ t[n, k], {n, 0, 9}, {k, n + 1}] // Flatten

Formula

T(n,k) = Sum_{j=0..n+1-k}binomial(n+1-k,j)*k^j*j^(n+1-k-j) for n>=0 and T(0,k) for k>=1.
E.g.f. of column k: exp(k*x*exp(x)).
With t(n,k) = T(n+k-1,k), t(n,k+j) = Sum_{i=0..n}binomial(n,i)*t(i,k)*t(n-i,j).

A356827 Expansion of e.g.f. exp(x * exp(3*x)).

Original entry on oeis.org

1, 1, 7, 46, 361, 3436, 37729, 463366, 6280369, 93015352, 1491337441, 25684077706, 472217487625, 9221588527204, 190441412508481, 4143470377262806, 94663498086222049, 2264440394856702832, 56570146384760433217, 1472545685988162638722
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Crossrefs

Programs

  • Maple
    A356827 := proc(n)
        add((3*k)^(n-k) * binomial(n,k),k=0..n) ;
    end proc:
    seq(A356827(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(3*x))))
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-3*k*x)^(k+1)))
    
  • PARI
    a(n) = sum(k=0, n, (3*k)^(n-k)*binomial(n, k));

Formula

G.f.: Sum_{k>=0} x^k / (1 - 3*k*x)^(k+1).
a(n) = Sum_{k=0..n} (3*k)^(n-k) * binomial(n,k).
Showing 1-3 of 3 results.