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A133121 Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1. 1
1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 2, 3, 1, 0, 1, 5, 4, 2, 2, 1, 0, 1, 6, 6, 3, 3, 2, 1, 0, 1, 8, 7, 5, 4, 2, 2, 1, 0, 1, 10, 8, 10, 3, 5, 2, 2, 1, 0, 1, 12, 13, 8, 9, 4, 4, 2, 2, 1, 0, 1, 15, 15, 14, 10, 8, 5, 4, 2, 2, 1, 0, 1, 18, 21, 15, 16, 8, 9, 4, 4, 2, 2, 1, 0, 1, 22, 25, 23, 17, 17, 7, 10, 4, 4, 2, 2, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

G.f.: Product_{n>=1} 1 + x^n/(1-y*x^n).

EXAMPLE

1

1,1

2,0,1

2,2,0,1

3,2,1,0,1

4,2,3,1,0,1

5,4,2,2,1,0,1

6,6,3,3,2,1,0,1

8,7,5,4,2,2,1,0,1

10,8,10,3,5,2,2,1,0,1

12,13,8,9,4,4,2,2,1,0,1

15,15,14,10,8,5,4,2,2,1,0,1

18,21,15,16,8,9,4,4,2,2,1,0,1

MAPLE

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,

       add(x^`if`(j=0, 0, j-1)*b(n-i*j, i-1), j=0..n/i))))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n$2)):

seq(T(n), n=1..16);  # Alois P. Heinz, Aug 21 2015

MATHEMATICA

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^If[j == 0, 0, j-1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[ Coefficient[p, x, i], {i, 0, n - 1}]][b[n, n]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-Fran├žois Alcover, Jan 23 2016, after Alois P. Heinz *)

PROG

(PARI) partitm(n, m, nmin)={ local(resul, partj) ; if( n < 0 || m <0, return([; ]) ; ) ; resul=matrix(0, m); if(m==0, return(resul); ) ; for(j=max(1, nmin), n\m, partj=partitm(n-j, m-1, j) ; for(r1=1, matsize(partj)[1], resul=concat(resul, concat([j], partj[r1, ])) ; ) ; ) ; if(m==1 && n >= nmin, resul=concat(resul, [[n]]) ; ) ; return(resul) ; }

partit(n)={ local(resul, partm, filr) ; if( n < 0, return([; ]) ; ) ; resul=matrix(0, n) ; for(m=1, n, partm=partitm(n, m, 1) ; filr=vector(n-m) ; for(r1=1, matsize(partm)[1], resul=concat( resul, concat(partm[r1, ], filr) ) ; ) ; ) ; return(resul) ; }

A133121row(n)={ local(p=partit(n), resul=vector(n), nprts, ndprts) ; for(r=1, matsize(p)[1], nprts=0 ; ndprts=0 ; for(c=1, n, if( p[r, c]==0, break, nprts++ ; if(c==1, ndprts++, if(p[r, c]!=p[r, c-1], ndprts++ ) ; ) ; ) ; ) ; k=nprts-ndprts; resul[k+1]++ ; ) ; return(resul) ; }

A133121()={ for(n=1, 20, arow=A133121row(n) ; for(k=1, n, print1(arow[k], ", ") ; ) ; ) ; }

A133121() ; \\ R. J. Mathar, Sep 28 2007

(PARI) tabl(nn) = my(pl = prod(n=1, nn, 1+x^n/(1-y*x^n)) + O(x^nn)); for (k=1, nn-1, print(Vecrev(polcoeff(pl, k, x)))); \\ Michel Marcus, Aug 23 2015

CROSSREFS

Cf. A000009, A090858.

Sequence in context: A113680 A128187 A266477 * A091602 A035465 A096144

Adjacent sequences:  A133118 A133119 A133120 * A133122 A133123 A133124

KEYWORD

easy,nonn,tabl

AUTHOR

Vladeta Jovovic, Sep 18 2007

EXTENSIONS

More terms from R. J. Mathar, Sep 28 2007

STATUS

approved

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Last modified June 26 13:27 EDT 2017. Contains 288766 sequences.