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A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k). 41

%I

%S 1,1,0,1,1,0,1,2,2,0,1,3,5,3,0,1,4,9,10,5,0,1,5,14,22,20,7,0,1,6,20,

%T 40,51,36,11,0,1,7,27,65,105,108,65,15,0,1,8,35,98,190,252,221,110,22,

%U 0,1,9,44,140,315,506,574,429,185,30,0,1,10,54,192,490,918,1265,1240,810,300,42,0

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

%C A(n,k) is also the number of partitions of n into parts of k kinds.

%C In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - _Vaclav Kotesovec_, Feb 28 2015, extended Jan 16 2017

%H Alois P. Heinz, <a href="/A144064/b144064.txt">Antidiagonals n = 0..140, flattened</a>

%H G. E. Bergum and V. E. Hoggatt, Jr., <a href="/A006503/a006503.pdf">Numerator polynomial coefficient array for the convolved Fibonacci sequence</a>, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)

%H G. E. Bergum and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/14-1/bergum.pdf">Numerator polynomial coefficient array for the convolved Fibonacci sequence</a>, Fib. Quart., 14 (1976), 43-48. See Table 1.

%H 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. 8.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>

%F G.f. of column k: Product_{j>=1} 1/(1-x^j)^k.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 2, 3, 4, 5, ...

%e 0, 2, 5, 9, 14, 20, ...

%e 0, 3, 10, 22, 40, 65, ...

%e 0, 5, 20, 51, 105, 190, ...

%e 0, 7, 36, 108, 252, 506, ...

%p 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->k)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

%t a[0, _] = 1; a[_, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* _Jean-Fran├žois Alcover_, Dec 06 2013 *)

%t 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[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-Fran├žois Alcover_, Mar 30 2015, after _Alois P. Heinz_ *)

%o (Julia) # DedekindEta is defined in A000594.

%o A144064Column(k, len) = DedekindEta(len, -k)

%o for n in 0:8 A144064Column(n, 6) |> println end # _Peter Luschny_, Mar 10 2018

%Y Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.

%Y Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).

%Y Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.

%Y Main diagonal gives A008485.

%Y Antidiagonal sums give A067687.

%Y Cf. A060642, A122768, A246935, A255961, A261718.

%K nonn,tabl

%O 0,8

%A _Alois P. Heinz_, Sep 09 2008

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Last modified July 17 16:58 EDT 2019. Contains 325107 sequences. (Running on oeis4.)