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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.

%I A340995 #17 Feb 14 2021 15:52:47
%S A340995 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,
%T A340995 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,
%U A340995 210,376,391,279,133,44,9,1,0,10,76,348,688,880,738,441,184,54,10,1
%N 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.
%H A340995 Alois P. Heinz, <a href="/A340995/b340995.txt">Rows n = 0..200, flattened</a>
%F A340995 T(n,k) = [x^n] (Sum_{j>=1} phi(j)*x^j)^k.
%e A340995 Triangle T(n,k) begins:
%e A340995   1;
%e A340995   0,  1;
%e A340995   0,  1,  1;
%e A340995   0,  2,  2,   1;
%e A340995   0,  2,  5,   3,   1;
%e A340995   0,  4,  8,   9,   4,   1;
%e A340995   0,  2, 16,  19,  14,   5,   1;
%e A340995   0,  6, 20,  42,  36,  20,   6,   1;
%e A340995   0,  4, 36,  72,  89,  60,  27,   7,   1;
%e A340995   0,  6, 44, 134, 184, 165,  92,  35,   8,  1;
%e A340995   0,  4, 68, 210, 376, 391, 279, 133,  44,  9,  1;
%e A340995   0, 10, 76, 348, 688, 880, 738, 441, 184, 54, 10, 1;
%e A340995   ...
%p A340995 T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p A340995       `if`(k=1, `if`(n=0, 0, numtheory[phi](n)), (q->
%p A340995        add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A340995     end:
%p A340995 seq(seq(T(n, k), k=0..n), n=0..12);
%t A340995 T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
%t A340995      If[k == 1, If[n == 0, 0, EulerPhi[n]], With[{q = Quotient[k, 2]},
%t A340995      Sum[T[j, q]*T[n - j, k - q], {j, 0, n}]]]];
%t A340995 Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 13 2021, after _Alois P. Heinz_ *)
%Y A340995 Columns k=0-2 give (offsets may differ): A000007, A000010, A065093.
%Y A340995 Row sums give A159929.
%Y A340995 T(2n,n) gives A340994.
%K A340995 nonn,tabl
%O A340995 0,8
%A A340995 _Alois P. Heinz_, Feb 01 2021