A122175
Triangle, read by rows, where T(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 7, 1, 1, 5, 20, 28, 11, 1, 1, 6, 35, 84, 66, 16, 1, 1, 7, 56, 210, 286, 136, 22, 1, 1, 8, 84, 462, 1001, 816, 253, 29, 1, 1, 9, 120, 924, 3003, 3876, 2024, 435, 37, 1, 1, 10, 165, 1716, 8008, 15504, 12650, 4495, 703, 46, 1, 1, 11, 220
Offset: 0
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 4, 1;
1, 4, 10, 7, 1;
1, 5, 20, 28, 11, 1;
1, 6, 35, 84, 66, 16, 1;
1, 7, 56, 210, 286, 136, 22, 1;
1, 8, 84, 462, 1001, 816, 253, 29, 1; ...
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Table[Binomial[(k(k+1))/2+n-k,n-k],{n,0,20},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 22 2016 *)
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T(n,k)=if(n
A121436
Matrix inverse of triangle A122176, where A122176(n,k) = C( k*(k+1)/2 + n-k + 1, n-k) for n>=k>=0.
Original entry on oeis.org
1, -2, 1, 3, -3, 1, -7, 9, -5, 1, 26, -37, 25, -8, 1, -141, 210, -155, 60, -12, 1, 1034, -1575, 1215, -516, 126, -17, 1, -9693, 14943, -11806, 5270, -1426, 238, -23, 1, 111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1, -1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1
Offset: 0
Triangle begins:
1;
-2, 1;
3, -3, 1;
-7, 9, -5, 1;
26, -37, 25, -8, 1;
-141, 210, -155, 60, -12, 1;
1034, -1575, 1215, -516, 126, -17, 1;
-9693, 14943, -11806, 5270, -1426, 238, -23, 1;
111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1;
-1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1; ...
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/* Matrix Inverse of A122176 */
{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial((c-1)*(c-2)/2+r,r-c)))); return((M^-1)[n+1,k+1])}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
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/* Obtain by G.F. */
{T(n,k)=polcoeff(1-sum(j=0, n-k-1, T(j+k,k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2+2)), n-k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
A121439
Matrix inverse of triangle A121334, where A121334(n,k) = C( n*(n+1)/2 + n-k, n-k) for n>=k>=0.
Original entry on oeis.org
1, -2, 1, -2, -4, 1, -14, 0, -7, 1, -143, -22, 11, -11, 1, -1928, -260, -40, 40, -16, 1, -32219, -3894, -385, -121, 99, -22, 1, -640784, -70644, -6496, -406, -406, 203, -29, 1, -14753528, -1502940, -128723, -9583, 259, -1184, 370, -37, 1, -385500056, -36631962, -2947266, -205620, -14076, 3657, -2967, 621
Offset: 0
Triangle, A121334^-1, begins:
1;
-2, 1;
-2, -4, 1;
-14, 0, -7, 1;
-143, -22, 11, -11, 1;
-1928, -260, -40, 40, -16, 1;
-32219, -3894, -385, -121, 99, -22, 1;
-640784, -70644, -6496, -406, -406, 203, -29, 1;
-14753528, -1502940, -128723, -9583, 259, -1184, 370, -37, 1; ...
Triangle A121412 begins:
1;
1, 1;
3, 1, 1;
18, 4, 1, 1;
170, 30, 5, 1, 1; ...
Row 3 of A121334^-1 equals row 3 of A121412^(-7), which begins:
1;
-7, 1;
7, -7, 1;
-14, 0, -7, 1; ...
Row 4 of A121334^-1 equals row 4 of A121412^(-11), which begins:
1;
-11, 1;
33, -11, 1;
-22, 22, -11, 1;
-143, -22, 11, -11, 1;...
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/* Matrix Inverse of A121334 */ {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(r*(r-1)/2+r-c,r-c)))); return((M^-1)[n+1,k+1])}
Showing 1-3 of 3 results.
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