A098545 -id:A098545 - cp's OEIS frontend

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.

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

Offset: 0

Alford Arnold, Sep 18 2006

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

			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.

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

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Green's relations
Wikipedia, Transformation semigroup

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

[(&+[Binomial(n,k)*StirlingSecond(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Feb 07 2019

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

Table[Sum[Binomial[n, k]*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]

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

a(n)=sum(k=0,n, binomial(n,k) * stirling(n,k,2) ); /* Joerg Arndt, Jun 16 2012 */

[sum(binomial(n,k)*stirling_number2(n,k) for k in (0..n)) for n in range(20)] # G. C. Greubel, Feb 07 2019

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

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

A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 6, 4, 1, 5, 10, 10, 10, 10, 5, 1, 6, 15, 15, 15, 20, 20, 20, 15, 15, 6, 1, 7, 21, 21, 21, 35, 35, 35, 35, 35, 35, 35, 21, 21, 7, 1, 8, 28, 28, 28, 28, 56, 56, 56, 56, 56, 70, 70, 70, 70, 70, 56, 56, 56, 28, 28, 8, 1, 9, 36, 36, 36, 36, 84, 84, 84, 84, 84, 84, 84

Offset: 1

Alford Arnold, Sep 14 2004

A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end since A098346 times A049019 also yields A049009. (cf. A000312 and A000670).

Partitions are in Abramowitz and Stegun order. - Franklin T. Adams-Watters, Nov 20 2006

			A036042 begins 1 2 2 3 3 3 4 4 4 4 4 ...
A036043 begins 1 1 2 1 2 3 1 2 2 3 4 ...
so a(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
Table begins
.
1
2 1
3 3  1
4 6  6  4  1
5 10 10 10 10 5  1
6 15 15 20 15 20 15 20 15 6 1
.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A090657, A000041 (row lengths), A098545 (row sums), A036036, A036042, A036043.

Table[Sequence @@
  Map[Function[p, Binomial[n, Length[p]]], IntegerPartitions[n]], {n,
  1, 10}] (* Olivier Gérard, May 07 2024 *)

a(n) = Combin( A036042(n), A036043(n) )

A179236 Irregular triangle T(n,k) = A096162(n,k)* A036040(n,k)* A048996(n,k)A098546(n,k) A178886(n,k) read by rows, 1<=k<=A000041(n).

Original entry on oeis.org

1, 2, 2, 6, 36, 6, 24, 192, 72, 432, 24, 120, 1200, 2400, 3600, 5400, 4800, 120, 720, 8640, 21600, 7200, 32400, 259200, 10800, 57600, 194400, 54000, 720, 5040, 70560, 211680, 352800, 317520, 3175200, 1058400, 1587600, 705600, 12700800, 2116800, 882000, 5292000, 635040, 5040, 40320, 645120, 2257920, 4515840

Offset: 1

Alford Arnold, Jul 04 2010

			The factor sequences begin
1..1..2..1..1..6
1..1..1..1..3..1
1..1..1..1..2..1
1..2..1..3..3..1
1..1..1..2..2..1
so the present sequence begins
1..2..2..6..36..6

Cf. A000041 (row lengths) A096161 A000110 A000079 A098545 A000522 A179235 (row sums)

cp's OEIS Frontend

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.

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A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

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A179236 Irregular triangle T(n,k) = A096162(n,k)* A036040(n,k)* A048996(n,k)A098546(n,k) A178886(n,k) read by rows, 1<=k<=A000041(n).

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cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A179236 Irregular triangle T(n,k) = A096162(n,k)* A036040(n,k)* A048996(n,k)*A098546(n,k)* A178886(n,k) read by rows, 1<=k<=A000041(n).

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A179236 Irregular triangle T(n,k) = A096162(n,k)* A036040(n,k)* A048996(n,k)A098546(n,k) A178886(n,k) read by rows, 1<=k<=A000041(n).