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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. 4
 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 From Gus Wiseman, Jan 23 2019: (Start) It is possible to augment the triangle to cover the n = 0 and k = n cases, giving:    1    1  0    1  1  0    2  0  1  0    2  2  0  1  0    3  2  1  0  1  0    4  2  3  1  0  1  0    5  4  2  2  1  0  1  0    6  6  3  3  2  1  0  1  0    8  7  5  4  2  2  1  0  1  0   10  8 10  3  5  2  2  1  0  1  0   12 13  8  9  4  4  2  2  1  0  1  0   15 15 14 10  8  5  4  2  2  1  0  1  0   18 21 15 16  8  9  4  4  2  2  1  0  1  0   22 25 23 17 17  7 10  4  4  2  2  1  0  1  0   27 30 32 21 19 16  8  9  4  4  2  2  1  0  1  0 Row seven {5, 4, 2, 2, 1, 0, 1, 0} counts the following integer partitions (empty columns not shown).   (7)    (322)   (2221)  (22111)  (211111)  (1111111)   (43)   (331)   (4111)  (31111)   (52)   (511)   (61)   (3211)   (421) (End) 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-François Alcover, Jan 23 2016, after Alois P. Heinz *) Table[Length[Select[IntegerPartitions[n], Length[#]-Length[Union[#]]==k&]], {n, 0, 15}, {k, 0, n}] (* augmented version, Gus Wiseman, Jan 23 2019 *) 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 Row sums are A000041. Row polynomials evaluated at -1 are A268498. Row polynomials evaluated at 2 are A006951. Cf. A000009, A090858, A100471, A116608. Sequence in context: A316723 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 October 20 10:00 EDT 2019. Contains 328257 sequences. (Running on oeis4.)