A117264
Self-convolution square-root of A117263.
Original entry on oeis.org
1, 1, 3, 29, 831, 69107, 16944055, 12387543565, 27116679815367, 177967005474840987, 3503280999913078429261, 206872249547698485286567247, 36647212198301159763279385189667
Offset: 0
-
{a(n)=local(A2=vector(n+1,m,sum(k=0,m-1,3^((m-k-1)*(m+k-2)/2))));Vec(Ser(A2)^(1/2))[n+1]}
A117262
Triangle T, read by rows, where matrix inverse T^-1 has -3^n in the secondary diagonal: [T^-1](n+1,n) = -3^n, with all 1's in the main diagonal and zeros elsewhere.
Original entry on oeis.org
1, 1, 1, 3, 3, 1, 27, 27, 9, 1, 729, 729, 243, 27, 1, 59049, 59049, 19683, 2187, 81, 1, 14348907, 14348907, 4782969, 531441, 19683, 243, 1, 10460353203, 10460353203, 3486784401, 387420489, 14348907, 177147, 729, 1
Offset: 0
Triangle T begins:
1;
1,1;
3,3,1;
27,27,9,1;
729,729,243,27,1;
59049,59049,19683,2187,81,1;
14348907,14348907,4782969,531441,19683,243,1;
10460353203,10460353203,3486784401,387420489,14348907,177147,729,1;
Matrix inverse T^-1 has -3^n in the 2nd diagonal:
1,
-1,1,
0,-3,1,
0,0,-9,1,
0,0,0,-27,1,
0,0,0,0,-81,1,
0,0,0,0,0,-243,1, ...
Cf.
A047656 (column 0),
A117263 (row sums); variants:
A117250 (p=q=2),
A117252 (p=q=3),
A117254 (p=q=4),
A117256 (p=q=5),
A117258 (p=2, q=4),
A117260 (p=-1, q=2),
A117265 (p=-2, q=2).
-
{T(n,k)=local(m=1,p=-1,q=3,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}
Original entry on oeis.org
1, 2, 5, 21, 169, 2705, 86561, 5539905, 709107841, 181531607297, 92944182936065, 95174843326530561, 194918079132734588929, 798384452127680876253185, 6540365431829961738266091521, 107157347235102093119751643480065, 3511331954199825387348021853554769921
Offset: 0
-
Table[Sum[2^((n(n-1))/2-(k(k-1))/2),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 05 2023 *)
-
a(n)=sum(k=0,n,2^((n-k)*(n+k-1)/2))
Showing 1-3 of 3 results.
Comments