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A119441 Distribution of A063834 in Abramowitz and Stegun order. 5
1, 2, 1, 3, 2, 1, 5, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 11, 7, 10, 9, 5, 6, 8, 3, 4, 2, 1, 15, 11, 14, 15, 7, 10, 9, 12, 5, 6, 8, 3, 4, 2, 1, 22, 15, 22, 21, 25, 11, 14, 15, 20, 18, 7, 10, 9, 12, 16, 5, 6, 8, 3, 4, 2, 1, 30, 22, 30, 33, 35, 15, 22, 21 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..74.

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].

FORMULA

T(n,k) = product_{p=1..A036043(n,k)} A000041(c), 1<=k<=A000041(n), where c are the parts in the k-th partition of n. - R. J. Mathar, Jul 12 2013

EXAMPLE

1;

2, 1;

3, 2, 1;

5, 3, 4, 2, 1;

7, 5, 6, 3, 4, 2, 1;

T(5,2) = 5 because the second partition of 5 is 1+4 and 4 can be repartitioned in 5 different ways.

T(5,3) = 6 because the third partition of 5 is 2+3, where the 2 can be partitioned in 2 ways (2, 1+1) and the 3 can be partitioned in 3 ways (3, 1+2, 1+1+1), 6=2*3.

T(5,4) = 3 because the fourth partition of 5 is 1+1+3 and 3 can be partitioned in 3 different ways.

MAPLE

# Compare two partitions (list) in AS order.

AScompare := proc(p1, p2)

    if nops(p1) > nops(p2) then

        return 1;

    elif nops(p1) < nops(p2) then

        return -1;

    else

        for i from 1 to nops(p1) do

            if op(i, p1) > op(i, p2) then

                return 1;

            elif op(i, p1) < op(i, p2) then

                return -1;

            end if;

        end do:

        return 0 ;

    end if;

end proc:

# Produce list of partitions in AS order

ASPrts := proc(n)

    local pi, insrt, p, ex ;

    pi := [] ;

    for p in combinat[partition](n) do

        insrt := 0 ;

        for ex from 1 to nops(pi) do

            if AScompare(p, op(ex, pi)) > 0 then

                insrt := ex ;

            end if;

        end do:

        if nops(pi) = 0 then

            pi := [p] ;

        elif insrt = 0 then

            pi := [p, op(pi)] ;

        elif insrt = nops(pi) then

            pi := [op(pi), p] ;

        else

            pi := [op(1..insrt, pi), p, op(insrt+1..nops(pi), pi)] ;

        end if;

    end do:

    return pi ;

end proc:

A119441 := proc(n, k)

    local pi, a, p ;

    pi := ASPrts(n)[k] ;

    a := 1 ;

    for p in pi do

        a := a*combinat[numbpart](p) ;

    end do:

    a ;

end proc:

for n from 1 to 10 do

    for k from 1 to A000041(n) do

        printf("%d, ", A119441(n, k)) ;

    end do:

    printf("\n") ;

end do: # R. J. Mathar, Jul 12 2013

CROSSREFS

Cf. A063834, A119442, A000041 (row lengths and also first column)

Sequence in context: A304100 A179314 A204927 * A322083 A058399 A209434

Adjacent sequences:  A119438 A119439 A119440 * A119442 A119443 A119444

KEYWORD

easy,nonn,tabf

AUTHOR

Alford Arnold, May 19 2006

STATUS

approved

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Last modified June 25 21:43 EDT 2019. Contains 324357 sequences. (Running on oeis4.)