The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A327631 Number T(n,k) of parts in all proper k-times partitions of n; triangle T(n,k), n >= 1, 0 <= k <= n-1, read by rows. 8
 1, 1, 2, 1, 5, 3, 1, 11, 21, 12, 1, 19, 61, 74, 30, 1, 34, 205, 461, 432, 144, 1, 53, 474, 1652, 2671, 2030, 588, 1, 85, 1246, 6795, 17487, 23133, 15262, 3984, 1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980, 1, 191, 6277, 69812, 360114, 1002835, 1606434, 1483181, 734272, 151080 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted. T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero. Row n is the inverse binomial transform of the n-th row of array A327618. LINKS Alois P. Heinz, Rows n = 1..200, flattened Wikipedia, Partition (number theory) FORMULA T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327618(n,i). T(n,n-1) = n * A327639(n,n-1) = n * A327643(n) for n >= 1. EXAMPLE T(4,0) = 1: 4 (1 part). T(4,1) = 11 = 2 + 2 + 3 + 4: 4-> 31 (2 parts) 4-> 22 (2 parts) 4-> 211 (3 parts) 4-> 1111 (4 parts) T(4,2) = 21 = 3 + 4 + 3 + 3 + 4 + 4: 4-> 31 -> 211 (3 parts) 4-> 31 -> 1111 (4 parts) 4-> 22 -> 112 (3 parts) 4-> 22 -> 211 (3 parts) 4-> 22 -> 1111 (4 parts) 4-> 211-> 1111 (4 parts) T(4,3) = 12 = 4 + 4 + 4: 4-> 31 -> 211 -> 1111 (4 parts) 4-> 22 -> 112 -> 1111 (4 parts) 4-> 22 -> 211 -> 1111 (4 parts) Triangle T(n,k) begins: 1; 1, 2; 1, 5, 3; 1, 11, 21, 12; 1, 19, 61, 74, 30; 1, 34, 205, 461, 432, 144; 1, 53, 474, 1652, 2671, 2030, 588; 1, 85, 1246, 6795, 17487, 23133, 15262, 3984; 1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980; ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+ (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]* b(n-i, min(n-i, i), k)))(b(i\$2, k-1)))) end: T:= (n, k)-> add(b(n\$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k): seq(seq(T(n, k), k=0..n-1), n=1..12); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]]*h[[2]]/ h[[1]]}][h[[1]]*b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]]; T[n_, k_] := Sum[b[n, n, i][[2]]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 1, 12}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 07 2020, after Alois P. Heinz *) CROSSREFS Columns k=0-2 give: A057427, -1+A006128(n), A328042. Row sums give A327648. T(n,floor(n/2)) gives A328041. Cf. A327618, A327632, A327639, A327643. Sequence in context: A210792 A105728 A120095 * A130197 A106513 A054446 Adjacent sequences: A327628 A327629 A327630 * A327632 A327633 A327634 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 19 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 10 23:32 EDT 2024. Contains 375059 sequences. (Running on oeis4.)