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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. 15
1, 1, 3, 13, 71, 456, 3337, 27203, 243203, 2357356, 24554426, 272908736, 3218032897, 40065665043, 524575892037, 7197724224361, 103188239447115, 1541604242708064, 23945078236133674, 385890657416861532, 6440420888899573136, 111132957321230896024 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..500

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

EXAMPLE

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.

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}]

PROG

(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 */

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

(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

CROSSREFS

Cf. A000041, A000110, A036040, A098545, A098546, A122454.

Cf. A134090, A048993 (S2).

Sequence in context: A198447 A318223 A162326 * A126390 A272428 A167894

Adjacent sequences:  A122452 A122453 A122454 * A122456 A122457 A122458

KEYWORD

easy,nonn

AUTHOR

Alford Arnold, Sep 18 2006

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

STATUS

approved

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Last modified July 26 20:17 EDT 2021. Contains 346294 sequences. (Running on oeis4.)