cp's OEIS Frontend

A126150 Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301. Second term 4 times first term.

%I A126150 #14 Feb 03 2025 13:45:18
%S A126150 1,1,4,1,6,24,36,24,6,96,384,636,744,636,384,96,2976,11904,20256,
%T A126150 26304,28536,26304,20256,11904,2976,151416,605664,1042056,1407024,
%U A126150 1650456,1736064,1650456,1407024,1042056,605664,151416,11449296,45797184
%N A126150 Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301. Second term 4 times first term.
%F A126150 Sum_{k=0..2n} (-1)^k*C(2n,k)*T(n,k) = (-6)^n.
%e A126150 Triangle begins:
%e A126150   1;
%e A126150   1, 4, 1;
%e A126150   6, 24, 36, 24, 6;
%e A126150   96, 384, 636, 744, 636, 384, 96;
%e A126150   2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, 2976;
%e A126150   151416, 605664, 1042056, 1407024, 1650456, 1736064, 1650456, 1407024, 1042056, 605664, 151416; ...
%e A126150 If we write the triangle like this:
%e A126150   .......................... ....1;
%e A126150   ................... ....1, ....4, ....1;
%e A126150   ............ ....6, ...24, ...36, ...24, ....6;
%e A126150   ..... ...96, ..384, ..636, ..744, ..636, ..384, ...96;
%e A126150   .2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, .2976;
%e A126150 then the first term in each row is the sum of the previous row:
%e A126150   2976 = 96 + 384 + 636 + 744 + 636 + 384 + 96
%e A126150 the next term is 4 times the first:
%e A126150   11904 = 4*2976,
%e A126150 and the remaining terms in each row are obtained by the rule illustrated by:
%e A126150   20256 = 2*11904 - 2976 - 6*96;
%e A126150   26304 = 2*20256 - 11904 - 6*384;
%e A126150   28536 = 2*26304 - 20256 - 6*636;
%e A126150   26304 = 2*28536 - 26304 - 6*744;
%e A126150   20256 = 2*26304 - 28536 - 6*636;
%e A126150   11904 = 2*20256 - 26304 - 6*384;
%e A126150   2976 = 2*11904 - 20256 - 6*96.
%e A126150 An alternate recurrence is illustrated by:
%e A126150   11904 = 2976 + 3*(96 + 384 + 636 + 744 + 636 + 384 + 96);
%e A126150   20256 = 11904 + 3*(384 + 636 + 744 + 636 + 384);
%e A126150   26304 = 20256 + 3*(636 + 744 + 636);
%e A126150   28536 = 26304 + 3*(744);
%e A126150 and then for k>n, T(n,k) = T(n,2n-k).
%o A126150 (PARI) T(n,k)=local(p=3);if(2*n<k || k<0,0,if(n==0 && k==0,1,if(k==0,sum(j=0,2*n-2,T(n-1,j)), if(k==1,(p+1)*T(n,0),if(k<=n,2*T(n,k-1)-T(n,k-2)-2*p*T(n-1,k-2),T(n,2*n-k))))))
%o A126150 (PARI) /* Alternate Recurrence: */ T(n,k)=local(p=3);if(2*n<k || k<0,0,if(n==0 && k==0,1,if(k==0,sum(j=0,2*n-2,T(n-1,j)), if(k<=n,T(n,k-1)+p*sum(j=k-1,2*n-1-k,T(n-1,j)),T(n,2*n-k)))))
%Y A126150 Cf. A126151 (column 0); diagonals: A126152, A126153; A126154; variants: A008301, A125053, A126155.
%K A126150 nonn,tabl
%O A126150 0,3
%A A126150 _Paul D. Hanna_, Dec 19 2006