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%I A144064 #77 Aug 04 2024 11:15:25
%S A144064 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 A144064 40,51,36,11,0,1,7,27,65,105,108,65,15,0,1,8,35,98,190,252,221,110,22,
%U A144064 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 A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).
%C A144064 A(n,k) is also the number of partitions of n into parts of k kinds.
%C A144064 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
%C A144064 When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - _Geoffrey Critzer_, Nov 11 2022
%H A144064 Alois P. Heinz, Antidiagonals n = 0..140, flattened
%H A144064 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)
%H A144064 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.
%H A144064 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.
%H A144064 N. J. A. Sloane, Transforms
%H A144064 Index entries for expansions of Product_{k >= 1} (1-x^k)^m
%F A144064 G.f. of column k: Product_{j>=1} 1/(1-x^j)^k.
%F A144064 A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):
%e A144064 Square array begins:
%e A144064 1, 1, 1, 1, 1, 1, ...
%e A144064 0, 1, 2, 3, 4, 5, ...
%e A144064 0, 2, 5, 9, 14, 20, ...
%e A144064 0, 3, 10, 22, 40, 65, ...
%e A144064 0, 5, 20, 51, 105, 190, ...
%e A144064 0, 7, 36, 108, 252, 506, ...
%p A144064 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 A144064 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 *)
%t A144064 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_ *)
%o A144064 (Julia) # DedekindEta is defined in A000594.
%o A144064 A144064Column(k, len) = DedekindEta(len, -k)
%o A144064 for n in 0:8 A144064Column(n, 6) |> println end # _Peter Luschny_, Mar 10 2018
%o A144064 (PARI) Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ _M. F. Hasler_, Aug 04 2024
%Y A144064 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 A144064 Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).
%Y A144064 Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.
%Y A144064 Main diagonal gives A008485.
%Y A144064 Antidiagonal sums give A067687.
%Y A144064 Cf. A060642, A122768, A246935, A255961, A261718.
%K A144064 nonn,tabl
%O A144064 0,8
%A A144064 _Alois P. Heinz_, Sep 09 2008
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