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 A008289 Triangle read by rows: Q(n,m) = number of partitions of n into m distinct parts, n>=1, m>=1. 64
 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 4, 3, 1, 4, 4, 1, 1, 5, 5, 1, 1, 5, 7, 2, 1, 6, 8, 3, 1, 6, 10, 5, 1, 7, 12, 6, 1, 1, 7, 14, 9, 1, 1, 8, 16, 11, 2, 1, 8, 19, 15, 3, 1, 9, 21, 18, 5, 1, 9, 24, 23, 7, 1, 10, 27, 27, 10, 1, 1, 10, 30, 34, 13, 1, 1, 11, 33, 39, 18, 2, 1, 11, 37 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Row n contains A003056(n) = floor((sqrt(8*n+1)-1)/2) terms (number of terms increases by one at each triangular number). - Michael Somos, Dec 04 2002 Row sums give A000009. Q(n,m) is the number of partitions of n whose greatest part is m and every number in {1,2,...,m} occurs as a part at least once. - Geoffrey Critzer, Nov 17 2011 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115. LINKS T. D. Noe and Alois P. Heinz, Rows n = 1..500 of triangle, flattened (first 200 rows from T. D. Noe) Eric Weisstein's World of Mathematics, Partition Function Q. FORMULA G.f.: Product_{n>0} (1 + y*x^n) = 1 + Sum_{n>0, k>0} Q(n, k) * x^n * y^k. - Michael Somos, Dec 04 2002 Q(n, k) = Q(n-k, k) + Q(n-k, k-1) for n>k>=1, with Q(1, 1)=1, Q(n, 0)=0 (n>=1). - Paul D. Hanna, Mar 04 2005 G.f.: Sum_{n>0, k>0} x^n * y^(k*(k+1)/2) / Product_{i=1..k} (1 - y^i). - Michael Somos, Jul 11 2017 Sum_{k>=0} k! * Q(n,k) = A032020(n). - Alois P. Heinz, Feb 25 2020 Q(n, m) = A008284(n - m*(m-1)/2, m) = A026820(n - m*(m+1)/2, m), using for the latter, the extension A026820(n, k) = A026820(n, n) = A000041(n), for every k >= n >= 0. - Álvar Ibeas, Jul 23 2020 EXAMPLE Q(8,3) = 2 since 8 can be written in 2 ways as sum of 3 distinct positive integers: 5+2+1 and 4+3+1. Triangle starts:   1;   1;   1,  1;   1,  1;   1,  2;   1,  2,  1;   1,  3,  1;   1,  3,  2;   1,  4,  3;   1,  4,  4,  1;   1,  5,  5,  1;   1,  5,  7,  2;   1,  6,  8,  3;   1,  6, 10,  5;   1,  7, 12,  6,  1;   1,  7, 14,  9,  1;   1,  8, 16, 11,  2;   1,  8, 19, 15,  3;   1,  9, 21, 18,  5;   1,  9, 24, 23,  7;   1, 10, 27, 27, 10,  1;   1, 10, 30, 34, 13,  1;   1, 11, 33, 39, 18,  2;   1, 11, 37, 47, 23,  3;   1, 12, 40, 54, 30,  5;   1, 12, 44, 64, 37,  7;   1, 13, 48, 72, 47, 11;   1, 13, 52, 84, 57, 14, 1;   1, 14, 56, 94, 70, 20, 1; ... Q(8,3) = 2 because there are 2 partitions of 8 in which  1, 2 and 3 occur as a part at least once: (3,2,2,1), (3,2,1,1,1). - Geoffrey Critzer, Nov 17 2011 MAPLE g:=product(1+t*x^j, j=1..40): gser:=simplify(series(g, x=0, 32)): P[0]:=1: for n from 1 to 30 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 25 do seq(coeff(P[n], t, j), j=1..floor((sqrt(8*n+1)-1)/2)) od; # yields sequence in triangular form; Emeric Deutsch, Feb 21 2006 # second Maple program: b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)       -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))     end: T:= n-> subsop(1=NULL, b(n, n))[]: seq(T(n), n=1..40);  # Alois P. Heinz, Nov 18 2012 MATHEMATICA q[n_, k_] := q[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]]; Take[ Flatten[ Table[q[n, k], {n, 1, 24}, {k, 1, Floor[(Sqrt[8n+1] - 1)/2]}]], 91] (* Jean-François Alcover, Aug 01 2011, after PARI prog. *) (* As a triangular table: *) Table[Coefficient[Series[Product[1+t    x^i, {i, n}], {x, 0, n}], x^n t^m], {n, 24}, {m, n}] (* Wouter Meeussen, Feb 22 2014 *) Table[Count[PowersRepresentations[n, k, 1], _?(Nor[MemberQ[#, 0], MemberQ[Differences@ #, 0]] &)], {n, 23}, {k, Floor[(Sqrt[8 n + 1] - 1)/2]}] // Flatten (* Michael De Vlieger, Jul 12 2017 *) nrows = 24; d=Table[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &], {n, nrows}] ; Flatten@Table[Table[Count[d[[n]], x_ /; Length[x] == m], {m, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, nrows}] (* Robert Price, Aug 17 2020 *) PROG (PARI) {Q(n, k) = if( k<0 || k>n, 0, polcoeff( polcoeff( prod(i=1, n, 1 + y*x^i, 1 + x * O(x^n)), n), k))}; /* Michael Somos, Dec 04 2002 */ (PARI) Q(n, k)=if(n(sqrtint(8*n+1)-1)\2, 0, u = n - k *(k+1)/2; polcoeff( 1 / prod(i=1, k, 1 - x^i, 1 + x*O(x^u)), u))}; /* Michael Somos, Jul 11 2017 */ CROSSREFS Cf. A030699, A104382, A117147, A209318. Sum of n-th row is A000009(n). Sum(Q(n,k)*k, k>=1) = A015723(n). A060016 is another version. Cf. A032020. Sequence in context: A075104 A253667 A233932 * A326625 A188884 A116679 Adjacent sequences:  A008286 A008287 A008288 * A008290 A008291 A008292 KEYWORD nonn,tabf,easy,nice,look AUTHOR EXTENSIONS Additional comments from Michael Somos, Dec 04 2002 Entry revised by N. J. A. Sloane, Nov 20 2006 STATUS approved

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Last modified April 10 06:54 EDT 2021. Contains 342843 sequences. (Running on oeis4.)