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A177747 Convolution of A008805 (triangular numbers repeated) with itself.

%I A177747 #32 Dec 19 2024 12:17:10
%S A177747 1,2,7,12,27,42,77,112,182,252,378,504,714,924,1254,1584,2079,2574,
%T A177747 3289,4004,5005,6006,7371,8736,10556,12376,14756,17136,20196,23256,
%U A177747 27132,31008,35853,40698,46683,52668,59983,67298,76153,85008,95634,106260,118910,131560,146510
%N A177747 Convolution of A008805 (triangular numbers repeated) with itself.
%H A177747 Bruno Berselli, <a href="/A177747/b177747.txt">Table of n, a(n) for n = 0..1000</a>
%H A177747 Brian Hopkins and Aram Tangboonduangjit, <a href="https://arxiv.org/abs/2412.11528">Water Cells in Compositions of 1s and 2s</a>, arXiv:2412.11528 [math.CO], 2024. See p. 3.
%H A177747 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-8,-2,12,-2,-8,3,2,-1).
%F A177747 Square (1 + x + 3x^2 + 3x^3 + 6x^4 + 6x^5 + ...)
%F A177747 G.f.: 1/((x+1)^4*(x-1)^6). [_Bruno Berselli_, Mar 23 2012]
%F A177747 a(n) = (n+5)*(2*n*(n+10)*(n^2+10*n+35)+5*(2*n*(n+10)+39)*(-1)^n+573)/3840. [_Bruno Berselli_, Mar 23 2012]
%e A177747 As a multiplication table array:
%e A177747 .
%e A177747 1, 1, 3, 3, 6,...
%e A177747 1, 1, 3, 3,......
%e A177747 3, 3, 9,.........
%e A177747 3, 3,............
%e A177747 6,...............
%e A177747 .
%e A177747 Then taking antidiagonal sums of terms, we obtain 1, (1 + 1) = 2, (3 + 1 + 3) = 7,  (3 + 3 + 3 + 3) = 12, (6, + 3 + 9 + 3 + 6) = 27, etc.
%t A177747 lst = CoefficientList[ Series[1/((1 - x) (1 - x^2)^2), {x, 0, 111}], x]; t[n_, k_] := lst[[n]] lst[[k]]; f[n_] := Sum[ t[n - m + 1, m], {m, n}]; Array[f, 45] (* _Robert G. Wilson v_, Dec 18 2010 *)
%t A177747 LinearRecurrence[{2, 3, -8, -2, 12, -2, -8, 3, 2, -1}, {1, 2, 7, 12, 27, 42, 77, 112, 182, 252}, 45] (* _Bruno Berselli_, Mar 23 2012 *)
%o A177747 (Magma) A008805:=func<i|(2*i^2+10*i+11+(2*i+5)*(-1)^i)/16>; [&+[A008805(i)*A008805(n-i): i in [0..n]]: n in [0..44]]; // _Bruno Berselli_, Mar 23 2012
%Y A177747 Cf. A008805.
%K A177747 nonn,easy
%O A177747 0,2
%A A177747 _Gary W. Adamson_, Dec 17 2010
%E A177747 More terms from _Robert G. Wilson v_, Dec 18 2010
%E A177747 Definition rewritten by _Bruno Berselli_, Mar 23 2012