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 A246609 Number T(n,k) of endofunctions on [n] whose cycle lengths are multiples of k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 11
 1, 0, 1, 0, 4, 1, 0, 27, 6, 2, 0, 256, 57, 24, 6, 0, 3125, 680, 300, 120, 24, 0, 46656, 9945, 4480, 2160, 720, 120, 0, 823543, 172032, 78750, 41160, 17640, 5040, 720, 0, 16777216, 3438673, 1591296, 866460, 430080, 161280, 40320, 5040 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k<=n. T(0,k) = 1, T(n,k) = 0 for k>n and n>0. Column k > 1 is asymptotic to n^(n - 1/2 + 1/(2*k)) * sqrt(2*Pi) / (2^(1/(2*k)) * k^(1/k) * GAMMA(1/(2*k))) * (1 - (3*k-1)*(k-1) * sqrt(2/n) * GAMMA(1/(2*k)) / (12 * k^2 * GAMMA(1/2+1/(2*k)))). - Vaclav Kotesovec, Sep 01 2014 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA E.g.f. for column k > 0: 1 / (1 - (-1)^k * LambertW(-x)^k)^(1/k). - Vaclav Kotesovec, Sep 01 2014 EXAMPLE Triangle T(n,k) begins:   1;   0,      1;   0,      4,      1;   0,     27,      6,     2;   0,    256,     57,    24,     6;   0,   3125,    680,   300,   120,    24;   0,  46656,   9945,  4480,  2160,   720,  120;   0, 823543, 172032, 78750, 41160, 17640, 5040, 720;   ... MAPLE with(combinat): b:= proc(n, i, k) option remember; `if`(n=0, 1,       `if`(i=0 or i>n, 0, add(b(n-i*j, i+k, k)*(i-1)!^j*       multinomial(n, n-i*j, i\$j)/j!, j=0..n/i)))     end: T:= (n, k)->add(b(j, k\$2)*n^(n-j)*binomial(n-1, j-1), j=0..n): seq(seq(T(n, k), k=0..n), n=0..10); MATHEMATICA multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0 || i > n, 0, Sum[b[n-i*j, i+k, k]*(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!, {j, 0, n/i}]]]; T[0, 0] = 1; T[n_, k_] := Sum[b[j, k, k]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A000312, A060435, A246610, A246611, A246612, A246613, A246614, A246615, A246616, A246617. Main diagonal gives A000142(n-1) for n>0. T(2n,n) gives A246618. Sequence in context: A081114 A069018 A156811 * A130636 A299354 A117414 Adjacent sequences:  A246606 A246607 A246608 * A246610 A246611 A246612 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 31 2014 STATUS approved

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Last modified January 19 22:12 EST 2022. Contains 350466 sequences. (Running on oeis4.)