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A007044 Left diagonal of partition triangle A047812.

%I A007044 M4370 #34 Nov 23 2020 06:06:59
%S A007044 0,0,1,7,20,48,100,194,352,615,1034,1693,2705,4239,6522,9889,14786,
%T A007044 21844,31913,46165,66162,94035,132600,185637,258128,356674,489906,
%U A007044 669173,909212,1229217,1653993,2215597,2955192,3925659,5194520,6847963,8995524,11776227
%N A007044 Left diagonal of partition triangle A047812.
%D A007044 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007044 Alois P. Heinz, <a href="/A007044/b007044.txt">Table of n, a(n) for n = 1..500</a>
%H A007044 R. K. Guy, <a href="/A007042/a007042.pdf">Letter to N. J. A. Sloane, Aug. 1992</a>
%H A007044 R. K. Guy, <a href="/A007042/a007042_1.pdf">Parker's permutation problem involves the Catalan numbers</a>, Preprint, 1992. (Annotated scanned copy)
%H A007044 R. K. Guy, <a href="http://www.jstor.org/stable/2324467">Parker's permutation problem involves the Catalan numbers</a>, Amer. Math. Monthly 100 (1993), 287-289.
%p A007044 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
%p A007044       or t*i<n, 0, b(n, i-1, t)+b(n-i, min(i, n-i), t-1)))
%p A007044     end:
%p A007044 a:= n-> b(2*n+2, n$2):
%p A007044 seq(a(n), n=1..50);  # _Alois P. Heinz_, May 31 2020
%t A007044 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
%t A007044 a[n_] := b[2n+2, n, n];
%t A007044 Array[a, 50] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)
%o A007044 (PARI) T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
%o A007044 for(n=1, 40, print1(T(n, 2), ",")) \\ _Petros Hadjicostas_, May 31 2020
%Y A007044 Cf. A007042, A007045, A047812, A051643.
%K A007044 nonn
%O A007044 1,4
%A A007044 _N. J. A. Sloane_ and _R. K. Guy_
%E A007044 Name edited by _Petros Hadjicostas_, May 31 2020