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A178440 Convolution square of A058187, the tetrahedral series with repeats.

%I A178440 #20 Dec 19 2024 10:49:11
%S A178440 1,2,9,16,44,72,156,240,450,660,1122,1584,2508,3432,5148,6864,9867,
%T A178440 12870,17875,22880,30888,38896,51272,63648,82212,100776,127908,155040,
%U A178440 193800,232560,286824,341088,415701,490314,591261,692208,826804,961400,1138500,1315600
%N A178440 Convolution square of A058187, the tetrahedral series with repeats.
%H A178440 Michael De Vlieger, <a href="/A178440/b178440.txt">Table of n, a(n) for n = 0..10000</a>
%H A178440 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 A178440 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2, 5, -12, -9, 30, 5, -40, 5, 30, -9, -12, 5, 2, -1).
%F A178440 Square (1 + x + 4x^2 + 4x^3 + 10x^4 + ...) = (1 + 2x + 9x^2 + ...).
%F A178440 G.f.: 1 / ( (1+x)^6*(x-1)^8 ). - _R. J. Mathar_, Jul 21 2015
%e A178440 Antidiagonal sums of terms in the array:
%e A178440   1,.. 1,. 4,. 4,. 10,  10,...
%e A178440   1,.. 1,. 4,. 4,. 10,........
%e A178440   4,.. 4,.16,.16,.............
%e A178440   4,.. 4,.16,.................
%e A178440   10,.10,.....................
%e A178440   10,.........................
%e A178440 Example: a(4) = 44 = (10 + 4 + 16 + 4 + 10).
%t A178440 LinearRecurrence[{2,5,-12,-9,30,5,-40,5,30,-9,-12,5,2,-1},{1,2,9,16,44,72,156,240,450,660,1122,1584,2508,3432},40] (* _Harvey P. Dale_, Apr 17 2020 *)
%Y A178440 Cf. A058187.
%K A178440 nonn,easy
%O A178440 0,2
%A A178440 _Gary W. Adamson_, Dec 22 2010
%E A178440 Corrected by _R. J. Mathar_, Jul 21 2015