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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). 42
 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, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS A(n,k) is also the number of partitions of n into parts of k kinds. 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 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy) G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1. Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8. N. J. A. Sloane, Transforms FORMULA G.f. of column k: Product_{j>=1} 1/(1-x^j)^k. A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i): EXAMPLE Square array begins:   1,   1,   1,   1,   1,   1, ...   0,   1,   2,   3,   4,   5, ...   0,   2,   5,   9,  14,  20, ...   0,   3,  10,  22,  40,  65, ...   0,   5,  20,  51, 105, 190, ...   0,   7,  36, 108, 252, 506, ... MAPLE 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); MATHEMATICA 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-François Alcover, Dec 06 2013 *) 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-François Alcover, Mar 30 2015, after Alois P. Heinz *) PROG (Julia) # DedekindEta is defined in A000594. A144064Column(k, len) = DedekindEta(len, -k) for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018 CROSSREFS 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. Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64). Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504. Main diagonal gives A008485. Antidiagonal sums give A067687. Cf. A060642, A122768, A246935, A255961, A261718. Sequence in context: A182888 A317205 A296068 * A172236 A191646 A297321 Adjacent sequences:  A144061 A144062 A144063 * A144065 A144066 A144067 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 09 2008 STATUS approved

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Last modified August 5 08:27 EDT 2021. Contains 346464 sequences. (Running on oeis4.)