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 A245501 Number A(n,k) of endofunctions f on [n] such that f^k(i) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 27, 1, 1, 1, 4, 10, 256, 1, 1, 1, 3, 19, 41, 3125, 1, 1, 1, 4, 12, 110, 196, 46656, 1, 1, 1, 3, 19, 73, 751, 1057, 823543, 1, 1, 1, 4, 10, 116, 556, 5902, 6322, 16777216, 1, 1, 1, 3, 21, 41, 901, 4737, 52165, 41393, 387420489, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(n,k) = n! * [x^n] exp(Sum_{d|(k-1)} (x*exp(x))^d/d) for k>1, A(n,0)=1, A(n,1)=n^n. EXAMPLE Square array A(n,k) begins:   1,     1,    1,    1,    1,    1,    1, ...   1,     1,    1,    1,    1,    1,    1, ...   1,     4,    3,    4,    3,    4,    3, ...   1,    27,   10,   19,   12,   19,   10, ...   1,   256,   41,  110,   73,  116,   41, ...   1,  3125,  196,  751,  556,  901,  220, ...   1, 46656, 1057, 5902, 4737, 8422, 1921, ... MAPLE with(numtheory): A:= (n, k)-> `if`(k=0, 1, `if`(k=1, n^n, n! *coeff(series(     exp(add((x*exp(x))^d/d, d=divisors(k-1))), x, n+1), x, n))): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA A[0, 1] = 1; A[n_, k_] := If[k==0, 1, If[k==1, n^n, n!*SeriesCoefficient[ Exp[ DivisorSum[k-1, (x*Exp[x])^#/#&]], {x, 0, n}]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *) CROSSREFS Column k=0-10 give: A000012, A000312, A000248, A060905, A060906, A060907, A245502, A245503, A245504, A245505, A245506. Main diagonal gives A245507. Sequence in context: A321592 A031278 A010328 * A247026 A193512 A276330 Adjacent sequences:  A245498 A245499 A245500 * A245502 A245503 A245504 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 24 2014 STATUS approved

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Last modified January 23 22:36 EST 2020. Contains 331177 sequences. (Running on oeis4.)