A098546 -id:A098546 - cp's OEIS frontend

A098545 Row sums of A098546.

1, 3, 7, 21, 51, 148, 365, 983, 2461, 6360, 15687, 39757, 97033, 240425, 582622, 1421273, 3409861, 8222920, 19565707, 46680362, 110309476, 260876036, 612293443, 1437616751, 3354111156, 7823501148, 18157700800, 42112132458

Offset: 1

Alford Arnold, Sep 14 2004

By using multisets (cf. A001700) and multinomials (cf. A005651); A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end (compare A090657).

			A098546 begins
1
1 2
1 3 3
1 4 6 6 4
so sequence begins 1 3 7 21 ...

Cf. A000041, A000312.

a(n) = Sum_{k=1..n} binomial(n, k)*A008284(n, k). - Vladeta Jovovic, Jul 24 2005

More terms from Vladeta Jovovic, Jul 24 2005

A122454 A triangle with shape A000041 defined by sequence A098546 times sequence A036040.

Original entry on oeis.org

1, 2, 1, 3, 9, 1, 4, 24, 18, 24, 1, 5, 50, 100, 100, 150, 50, 1, 6, 90, 225, 150, 300, 1200, 300, 300, 675, 90, 1, 7, 147, 441, 735, 735, 3675, 2450, 3675, 1225, 7350, 3675, 735, 2205, 147, 1, 8, 224, 784, 1568, 980, 1568, 9408, 15680, 11760, 15680, 3920, 29400

Offset: 1

Alford Arnold, Sep 18 2006

Shape sequence for A122454 is A000041 which counts numeric partitions.

			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 the triangle begins:
1;
2,   1;
3,   9,   1;
4,  24,  18,  24,   1;
5,  50, 100, 100, 150,   50,    1;
6,  90, 225, 150, 300, 1200,  300,  300,  675,   90,    1;
7, 147, 441, 735, 735, 3675, 2450, 3675, 1225, 7350, 3675, 735, 2205, 147, 1;

Cf. A122455.

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: for n from 1 to 10 do for k from 1 to combinat[numbpart](n) do a:=A122454(n,k) ; printf("%d, ",a) ; od; od ; # R. J. Mathar, Jul 17 2007

A122454(n) = A098546(n) times A036040(n).

More terms from R. J. Mathar, Jul 17 2007

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)

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

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

A181415 Irregular triangle a(n,k) = A049009(n,k)/n, read by rows 1<=k<=A000041(n).

Original entry on oeis.org

1, 1, 1, 1, 6, 2, 1, 12, 9, 36, 6, 1, 20, 40, 120, 180, 240, 24, 1, 30, 75, 50, 300, 1200, 300, 1200, 2700, 1800, 120, 1, 42, 126, 210, 630, 3150, 2100, 3150, 4200, 25200, 12600, 12600, 37800, 15120, 720, 1, 56, 196, 392, 245, 1176, 7056, 11760, 8820, 11760, 11760, 88200

Offset: 1

Alford Arnold, Oct 20 2010

			Row three is calculated as follows:
( 3 18 6) divided by (3 3 3) yielding (1 6 2)
1;
1,1;
1,6,2;
1,12,9,36,6;
1,20,40,120,180,240,24;
1,30,75,50,300,1200,300,1200,2700,1800,120;
1,42,126,210,630,3150,2100,3150,4200,25200,12600,12600,37800,15120,720;

Cf. A000169 (row sums), A000081 (unlabeled rooted trees) A179438 (a similar refinement), A054589, A135278, A019538, A101817, A101818

Sum_{k=1.. A000041(n)} a(n,k) = A000169(n). (Row sums)

a(n,k) = A098546(n,k) *A049019(n,k) /n. - Compare with the formula in A101818.

Edited by R. J. Mathar, May 17 2016

cp's OEIS Frontend

A098545 Row sums of A098546.

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A122454 A triangle with shape A000041 defined by sequence A098546 times sequence A036040.

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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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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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A181415 Irregular triangle a(n,k) = A049009(n,k)/n, read by rows 1<=k<=A000041(n).

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Formula

Extensions

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).