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A144048 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j^k).

%I A144048 #36 Sep 23 2022 16:13:04
%S A144048 1,1,1,1,1,2,1,1,3,3,1,1,5,6,5,1,1,9,14,13,7,1,1,17,36,40,24,11,1,1,
%T A144048 33,98,136,101,48,15,1,1,65,276,490,477,266,86,22,1,1,129,794,1828,
%U A144048 2411,1703,649,160,30,1,1,257,2316,6970,12729,11940,5746,1593,282,42,1,1,513
%N A144048 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j^k).
%C A144048 In general, column k > 0 is asymptotic to (Gamma(k+2)*Zeta(k+2))^((1-2*Zeta(-k)) /(2*k+4)) * exp((k+2)/(k+1) * (Gamma(k+2)*Zeta(k+2))^(1/(k+2)) * n^((k+1)/(k+2)) + Zeta'(-k)) / (sqrt(2*Pi*(k+2)) * n^((k+3-2*Zeta(-k))/(2*k+4))). - _Vaclav Kotesovec_, Mar 01 2015
%H A144048 Alois P. Heinz, <a href="/A144048/b144048.txt">Antidiagonals = 0..99, flattened</a>
%H A144048 Vaclav Kotesovec, <a href="https://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.
%H A144048 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A144048 G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j^k).
%e A144048 Square array begins:
%e A144048   1,  1,   1,   1,    1,     1, ...
%e A144048   1,  1,   1,   1,    1,     1, ...
%e A144048   2,  3,   5,   9,   17,    33, ...
%e A144048   3,  6,  14,  36,   98,   276, ...
%e A144048   5, 13,  40, 136,  490,  1828, ...
%e A144048   7, 24, 101, 477, 2411, 12729, ...
%p A144048 with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->j^k)(n); seq(seq(A(n,d-n), n=0..d), d=0..13);
%t A144048 etr[p_] := Module[{ b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[j, j^k]][n]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 13}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A144048 Columns k=0-9 give: A000041, A000219, A023871, A023872, A023873, A023874, A023875, A023876, A023877, A023878.
%Y A144048 Rows give: 0-1: A000012, 2: A000051, A094373, 3: A001550, 4: A283456, 5: A283457.
%Y A144048 Main diagonal gives A252782.
%Y A144048 Cf. A283272.
%K A144048 nonn,tabl
%O A144048 0,6
%A A144048 _Alois P. Heinz_, Sep 08 2008