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(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))))))
(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)))))
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(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))))))} (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)))))}
(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)))))
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_Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 19 2006
20256 = 2*11904 - 2976 - 6*96 ;
26304 = 2*20256 - 11904 - 6*384 ;
28536 = 2*26304 - 20256 - 6*636 ;
26304 = 2*28536 - 26304 - 6*744 ;
20256 = 2*26304 - 28536 - 6*636 ;
11904 = 2*20256 - 26304 - 6*384 ;
nonn,tabl,new
nonn,tabl,new
Paul D . Hanna (pauldhanna(AT)juno.com), Dec 19 2006
2976 = 2*11904 - 20256 - 6*96 .
nonn,tabl,new
Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301.
1, 1, 4, 1, 6, 24, 36, 24, 6, 96, 384, 636, 744, 636, 384, 96, 2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, 2976, 151416, 605664, 1042056, 1407024, 1650456, 1736064, 1650456, 1407024, 1042056, 605664, 151416, 11449296, 45797184
0,3
Sum_{k=0,2n} (-1)^k*C(2n,k)*T(n,k) = (-6)^n.
Triangle begins:
1;
1, 4, 1;
6, 24, 36, 24, 6;
96, 384, 636, 744, 636, 384, 96;
2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, 2976;
151416, 605664, 1042056, 1407024, 1650456, 1736064, 1650456, 1407024, 1042056, 605664, 151416; ...
If we write the triangle like this:
.......................... ....1;
................... ....1, ....4, ....1;
............ ....6, ...24, ...36, ...24, ....6;
..... ...96, ..384, ..636, ..744, ..636, ..384, ...96;
.2976, 11904, 20256, 26304, 28536, 26304, 20256, 11904, .2976;
then the first term in each row is the sum of the previous row:
2976 = 96 + 384 + 636 + 744 + 636 + 384 + 96
the next term is 4 times the first:
11904 = 4*2976,
and the remaining terms in each row are obtained by the rule
illustrated by:
20256 = 2*11904 - 2976 - 6*96 ;
26304 = 2*20256 - 11904 - 6*384 ;
28536 = 2*26304 - 20256 - 6*636 ;
26304 = 2*28536 - 26304 - 6*744 ;
20256 = 2*26304 - 28536 - 6*636 ;
11904 = 2*20256 - 26304 - 6*384 ;
2976 = 2*11904 - 20256 - 6*96 .
An alternate recurrence is illustrated by:
11904 = 2976 + 3*(96 + 384 + 636 + 744 + 636 + 384 + 96);
20256 = 11904 + 3*(384 + 636 + 744 + 636 + 384);
26304 = 20256 + 3*(636 + 744 + 636);
28536 = 26304 + 3*(744);
and then for k>n, T(n,k) = T(n,2n-k).
(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))))))} (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)))))}
nonn,tabl
Paul D Hanna (pauldhanna(AT)juno.com), Dec 19 2006
approved