%I #27 Jan 14 2024 00:14:48
%S 1,415,8623,64405,289981,965071,2626975,6195673,13123945,25572511,
%T 46610191,80439085,132644773,210471535,323122591,482085361,701481745,
%U 998443423,1393512175,1911065221,2579765581,3433037455,4509566623,5853825865,7516625401,9555688351
%N a(n) is the number of semimagic 4 X 3 matrices whose entries are nonnegative integers, with row sum = 3*n and column sum = 4*n.
%C This sequence was found during Doron Zeilberger's Experimental Mathematics class on Feb. 11, 2019. All class members contributed to the discovery of this sequence.
%H Andrew Howroyd, <a href="/A306381/b306381.txt">Table of n, a(n) for n = 0..1000</a>
%H Doron Zeilberger, <a href="http://sites.math.rutgers.edu/~zeilberg/EM19/C6.txt">Experimental Mathematics Class in Spring 2019</a>, C6.txt; <a href="/A306381/a306381.txt">Local copy</a>
%F a(n) = (139/4)*n^6 + (417/4)*n^5 + (535/4)*n^4 + (375/4)*n^3 + (77/2)*n^2 + 9*n + 1.
%e For n=1 the a(1)= 415 because there are 415 4 X 3 matrices with nonnegative integer entries, whose row sum is 3 and column sum is 4.
%o (PARI) a(n) = {(139*n^6 + 417*n^5 + 535*n^4 + 375*n^3 + 154*n^2 + 36*n + 4)/4} \\ _Andrew Howroyd_, Mar 01 2020
%K nonn
%O 0,2
%A _Yukun Yao_, Feb 11 2019
%E a(0)=1 prepended and terms a(11) and beyond from _Andrew Howroyd_, Mar 01 2020
