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 A340995 Triangle T(n,k) whose k-th column is the k-fold self-convolution of the Euler totient function phi; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 4
 1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 4, 8, 9, 4, 1, 0, 2, 16, 19, 14, 5, 1, 0, 6, 20, 42, 36, 20, 6, 1, 0, 4, 36, 72, 89, 60, 27, 7, 1, 0, 6, 44, 134, 184, 165, 92, 35, 8, 1, 0, 4, 68, 210, 376, 391, 279, 133, 44, 9, 1, 0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Rows n = 0..200, flattened FORMULA T(n,k) = [x^n] (Sum_{j>=1} phi(j)*x^j)^k. EXAMPLE Triangle T(n,k) begins: 1; 0, 1; 0, 1, 1; 0, 2, 2, 1; 0, 2, 5, 3, 1; 0, 4, 8, 9, 4, 1; 0, 2, 16, 19, 14, 5, 1; 0, 6, 20, 42, 36, 20, 6, 1; 0, 4, 36, 72, 89, 60, 27, 7, 1; 0, 6, 44, 134, 184, 165, 92, 35, 8, 1; 0, 4, 68, 210, 376, 391, 279, 133, 44, 9, 1; 0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1; ... MAPLE T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, numtheory[phi](n)), (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2)))) end: seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0], If[k == 1, If[n == 0, 0, EulerPhi[n]], With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n - j, k - q], {j, 0, n}]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *) CROSSREFS Columns k=0-2 give (offsets may differ): A000007, A000010, A065093. Row sums give A159929. T(2n,n) gives A340994. Sequence in context: A060086 A308680 A177975 * A363733 A062135 A190182 Adjacent sequences: A340992 A340993 A340994 * A340996 A340997 A340998 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Feb 01 2021 STATUS approved

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Last modified April 17 04:44 EDT 2024. Contains 371756 sequences. (Running on oeis4.)