A102321
Column 0 of triangular matrix A102320, which satisfies T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).
Original entry on oeis.org
1, 1, 4, 33, 436, 8122, 197920, 6007205, 219413116, 9402081718, 463548752912, 25893783163498, 1618536618626888, 112053082721454708, 8518619080226661504, 705977323976245345133, 63382036275445226941548
Offset: 0
G.f.: 1 = (1-x) + 1*x*(1-x)(1-3x) + 4*x^2*(1-x)(1-3x)(1-5x) + ... + a(n)*x^n*(1-x)(1-3x)(1-5x)*..*(1-(2n+1)*x) + ...
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{a(n)=local(A=Mat(1),B); for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=2*j-1,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,1])}
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{a(n)=if(n==0,1,polcoeff(1-sum(k=0,n-1,a(k)*x^k*prod(j=0,k,1-(2*j+1)*x+x*O(x^n))),n))}
A102322
Column 1 of triangular matrix A102320, which that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).
Original entry on oeis.org
0, 3, 9, 72, 945, 17568, 427770, 12979080, 473981445, 20308813128, 1001231706582, 55927084380552, 3495759750651978, 242012640619081152, 18398411206663695732, 1524754064472700613328, 136890662566189661556525
Offset: 0
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{a(n)=local(A=Mat([1,0;1,1]),B); for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=2*j-1,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,2])}
A102323
Triangle, read by rows, where T(n,k) = T(n,k-1) + (2*k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.
Original entry on oeis.org
1, 1, 1, 1, 4, 4, 1, 13, 33, 33, 1, 40, 205, 436, 436, 1, 121, 1146, 4198, 8122, 8122, 1, 364, 6094, 35480, 108578, 197920, 197920, 1, 1093, 31563, 279923, 1257125, 3434245, 6007205, 6007205, 1, 3280, 161095, 2120556, 13434681, 51211376
Offset: 0
T(5,2) = 1146 = 1*1 + 3*40 + 5*205 = 1*T(4,0) + 3*T(4,1) + 5*T(4,2).
T(5,2) = 1146 = 121 + 5*205 = T(5,1) + (2*2+1)*T(4,2).
T(5,3) = 4198 = 1146 + 7*436 = T(5,2) + (2*3+1)*T(4,3).
Rows begin:
[1],
[1,1],
[1,4,4],
[1,13,33,33],
[1,40,205,436,436],
[1,121,1146,4198,8122,8122],
[1,364,6094,35480,108578,197920,197920],
[1,1093,31563,279923,1257125,3434245,6007205,6007205],...
Showing 1-3 of 3 results.
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