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 A222730 Total sum T(n,k) of parts <= n of multiplicity k in all partitions of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 12
 0, 0, 1, 3, 2, 1, 11, 6, 0, 1, 36, 10, 3, 0, 1, 79, 21, 3, 1, 0, 1, 186, 33, 7, 3, 1, 0, 1, 345, 59, 9, 4, 1, 1, 0, 1, 672, 89, 20, 4, 4, 1, 1, 0, 1, 1163, 145, 22, 11, 4, 2, 1, 1, 0, 1, 2026, 212, 44, 13, 6, 4, 2, 1, 1, 0, 1, 3273, 325, 56, 21, 8, 6, 2, 2, 1, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS For k > 0, column k is asymptotic to sqrt(3) * (2*k+1) * exp(Pi*sqrt(2*n/3)) / (2 * k^2 * (k+1)^2 * Pi^2) ~ 6 * (2*k+1) * n * p(n) / (k^2 * (k+1)^2 * Pi^2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, May 29 2018 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA Sum_{k=0..n} k*T(n,k) = A066186(n) = n*A000041(n). Sum_{k=1..n} T(n,k) = A014153(n-1) for n>0. Sum_{k=0..n} T(n,k) = n*(n+1)/2*A000041(n) = A000217(n)*A000041(n). (2 * Sum_{k=0..n} T(n,k)) / (Sum_{k=0..n} k*T(n,k)) = n+1 for n>0. T(2*n+1,n+1) = A002865(n). EXAMPLE The partitions of n=4 are [1,1,1,1], [2,1,1], [2,2], [3,1], [4].  Parts <= 4 with multiplicity m=0 sum up to (2+3+4)+(3+4)+(1+3+4)+(2+4)+(1+2+3) = 36, for m=1 the sum is 2+(3+1)+4 = 10, for m=2 the sum is 1+2 = 3, for m=3 the sum is 0, for m=4 the sum is 1 => row 4 = [36, 10, 3, 0, 1]. Triangle T(n,k) begins:     0;     0,  1;     3,  2,  1;    11,  6,  0, 1;    36, 10,  3, 0, 1;    79, 21,  3, 1, 0, 1;   186, 33,  7, 3, 1, 0, 1;   345, 59,  9, 4, 1, 1, 0, 1;   672, 89, 20, 4, 4, 1, 1, 0, 1; MAPLE b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0],       `if`(p=0, [0\$(n+2)], add((l-> subsop(m+2=p*l[1]+l[m+2], l))           ([b(n-p*m, p-1)[], 0\$(p*m)]), m=0..n/p)))     end: T:= n-> subsop(1=NULL, b(n, n))[]: seq(T(n), n=0..14); MATHEMATICA b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n - p*m, p-1] , Array[0&, p*m]]], {m, 0, n/p}]]]; Rest /@ Table[b[n, n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *) CROSSREFS Columns k=0-10 give: A213679, A103628, A117525, A222731, A222732, A222733, A222734, A222735, A222736, A222737, A222738. Cf. A000041, A000217, A002865, A014153, A066186. Sequence in context: A309951 A077756 A115080 * A104219 A123513 A117442 Adjacent sequences:  A222727 A222728 A222729 * A222731 A222732 A222733 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 03 2013 STATUS approved

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Last modified February 28 23:19 EST 2020. Contains 332353 sequences. (Running on oeis4.)