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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.

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A122455 a(n) = Sum_{k=0..n} C(n,k)*S2(n,k). Binomial convolution of the Stirling numbers of the 2nd kind. Also sum of the rows of A122454.

Original entry on oeis.org

1, 1, 3, 13, 71, 456, 3337, 27203, 243203, 2357356, 24554426, 272908736, 3218032897, 40065665043, 524575892037, 7197724224361, 103188239447115, 1541604242708064, 23945078236133674, 385890657416861532, 6440420888899573136, 111132957321230896024
Offset: 0

Views

Author

Alford Arnold, Sep 18 2006

Keywords

Comments

A122454(n,k) = A098546(n,k) times A036040(n,k) (two triangles shaped by integer partitions A000041(n)).
Row sums of A098546 give sequence A098545 and row sums of A036040 give sequence A000110 (the Bell numbers)
Equals column zero of triangle A134090: let C equal Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere; then a(n) = column 0 of row n of (I + D*C)^n (see A134090). - Paul D. Hanna, Oct 07 2007
Number of Green's H-classes in the full transformation semigroup on [n]. Row sums of A090683. - Geoffrey Critzer, Dec 27 2022

Examples

			A098546(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
A036040(n) begins 1 1 1 1 3 1 1 4 3 6 1 ...
so
A122454(n) begins 1 2 1 3 9 1 4 24 18 24 1 ...
and
the present sequence begins 1 3 13 71 ...
with A000041 entries per row.

References

  • O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, pages 58-62.

Links

Crossrefs

Cf. A000041, A000110, A036040, A098545, A098546, A122454.
Cf. A134090, A048993 (S2).
Cf. A090683.

Programs

  • Magma
    [(&+[Binomial(n,k)*StirlingSecond(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Feb 07 2019
  • Maple
    sortAbrSteg := proc(L1,L2) local i ; if nops(L1) < nops(L2) then RETURN(true) ; elif nops(L2) < nops(L1) then RETURN(false) ; else for i from 1 to nops(L1) do if op(i,L1) < op(i,L2) then RETURN(false) ; fi ; od ; RETURN(true) ; fi ; end: A098546 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then m := nops(op(k,prts)) ; binomial(n,m) ; else 0 ; fi ; end: M3 := proc(L) local n,k,an,resul; n := add(i,i=L) ; resul := factorial(n) ; for k from 1 to n do an := add(1-min(abs(j-k),1),j=L) ; resul := resul/ (factorial(k))^an /factorial(an) ; od ; end: A036040 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then M3(op(k,prts)) ; else 0 ; fi ; end: A122454 := proc(n,k) A098546(n,k)*A036040(n,k) ; end: A122455 := proc(n) add(A122454(n,k),k=1..combinat[numbpart](n)) ; end: seq(A122455(n),n=1..18) ; # R. J. Mathar, Jul 17 2007
# Alternatively:
A122455 := n -> add(binomial(n,k)*Stirling2(n,k),k=0..n):
seq(A122455(n),n=0..21); # Peter Luschny, Aug 11 2015
  • Mathematica
    Table[Sum[Binomial[n, k]*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n)= polcoeff(sum(k=0,n,binomial(n,k)*x^k/prod(i=0,k,1-i*x +x*O(x^n))),n) \\ Paul D. Hanna, Oct 07 2007
  • PARI
    a(n)=sum(k=0,n, binomial(n,k) * stirling(n,k,2) ); /* Joerg Arndt, Jun 16 2012 */
  • Sage
    [sum(binomial(n,k)*stirling_number2(n,k) for k in (0..n)) for n in range(20)] # G. C. Greubel, Feb 07 2019

Formula

a(n) = [x^n] Sum_{k=0..n} C(n,k) * x^k / [Product_{i=0..k} (1 - i*x)]; equivalently, a(n) = Sum_{k=0..n} C(n,k) * S2(n,k), where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind. - Paul D. Hanna, Oct 07 2007

Extensions

More terms from R. J. Mathar, Jul 17 2007
Definition modified by Olivier Gérard, Oct 23 2012
a(0)=1 prepended by Alois P. Heinz, Sep 17 2017

A363849 Triangular array read by rows. T(n,k) is the number of Green's H-classes of rank k in the semigroup of partial transformations, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 18, 1, 1, 60, 150, 40, 1, 1, 155, 900, 650, 75, 1, 1, 378, 4515, 7000, 2100, 126, 1, 1, 889, 20286, 59535, 36750, 5586, 196, 1, 1, 2040, 84700, 435120, 486570, 148176, 12936, 288, 1, 1, 4599, 335880, 2864820, 5358150, 2876202, 493920, 27000, 405, 1
Offset: 0

Views

Author

Geoffrey Critzer, Jun 24 2023

Keywords

Comments

Let H_f denote the H-class in the semigroup of partial transformations containing f. Then H_f contains an idempotent iff the image of f is a transversal for the kernel of f.
Let H_f ~ H_g iff the image of f is contained in the image of g and the kernel of f is more coarse than the kernel of g. Then ~ is a partial order on the H-classes, hence a preorder (quasi-order) on the semigroup. The poset is isomorphic to the Segre product of the Boolean lattice of rank n and the partition lattice of [n+1].

Examples

			Triangle begins:
 1;
 1,   1;
 1,   6,   1;
 1,  21,  18,   1;
 1,  60, 150,  40,  1;
 1, 155, 900, 650, 75, 1;
 ...

References

  • O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, 2009, Chapter 4.4 - 4.6.

Links

Crossrefs

Columns k=0-1 give: A000012, A066524.
Row sums give A134055(n+1).
T(n,n-1) gives A002411.
Cf. A000169, A007318, A008277, A090683, A101818.

Programs

  • Maple
    T:= (n, k)-> binomial(n, k)*Stirling2(n+1, k+1):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2023
  • Mathematica
    Table[Table[Binomial[n, k] StirlingS2[n + 1, k + 1], {k, 0, n}], {n,0, 5}] // Grid

Formula

T(n,k) = A007318(n,k)*A008277(n+1,k+1).
Sum_{k=0..n} T(n,k)*k! = (n+1)^n = A000169(n+1).
T(n,1) = A101818(n,1) = A066524(n) = n*(2^n - 1). (Every partial function of rank 1 is idempotent.)
Showing 1-2 of 2 results.

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