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 A327646 Total number of steps in all proper many times partitions of n. 2
 0, 0, 1, 4, 25, 108, 788, 4740, 44445, 339632, 3625136, 35508536, 462626736, 5273725108, 76634997096, 1047347436984, 17542238923677, 268193251446228, 4949536256552648, 86303019303031400, 1768833677916545596, 34165810747993948664, 759192269597947084836 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..400 FORMULA a(n) = Sum_{k=1..n-1} k * A327639(n,k). EXAMPLE a(3) = 4 = 0+1+1+2, counting steps "->" in: 3, 3->21, 3->111, 3->21->111. a(4) = 25: 4, 4->31, 4->22, 4->211, 4->1111, 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111, 4->31->211->1111, 4->22->112->1111, 4->22->211->1111. MAPLE b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,       b(n, i-1, k), 0) +b(i\$2, k-1)*b(n-i, min(n-i, i), k))     end: a:= n-> add(k*add(b(n\$2, i)*(-1)^(k-i)*         binomial(k, i), i=0..k), k=1..n-1): seq(a(n), n=0..23); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]]; a[n_] := Sum[k Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}], {k, 1, n - 1}]; a /@ Range[0, 23] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *) CROSSREFS Cf. A327639. Sequence in context: A042651 A225692 A070764 * A244746 A110051 A334551 Adjacent sequences:  A327643 A327644 A327645 * A327647 A327648 A327649 KEYWORD nonn AUTHOR Alois P. Heinz, Sep 20 2019 STATUS approved

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Last modified April 15 01:19 EDT 2021. Contains 342971 sequences. (Running on oeis4.)