A102920
Triangular matrix, read by rows, equal to the matrix square of A102098.
Original entry on oeis.org
1, 3, 4, 36, 40, 9, 1036, 1128, 189, 16, 56355, 61120, 9720, 576, 25, 5045370, 5466320, 857466, 47040, 1375, 36, 679409158, 735847800, 114915375, 6155008, 163500, 2808, 49, 129195427716, 139910204080, 21813099606, 1158059520, 29767000, 458136
Offset: 0
Rows begin:
[1],
[3,4],
[36,40,9],
[1036,1128,189,16],
[56355,61120,9720,576,25],
[5045370,5466320,857466,47040,1375,36],
[679409158,735847800,114915375,6155008,163500,2808,49],...
-
{T(n,k)=local(A=matrix(1,1),B);A[1,1]=1; for (m=2,n+1,B=matrix(m,m);for (i=1,m, for (j=1,i, if(j==i,B[i,j]=j,B[i,j]=(A^3)[i-1,j]);));A=B); return((A^2)[n+1,k+1])}
A102916
Triangle, read by rows, where the antidiagonals are formed by interleaving the rows of triangle A102098 with the rows of its matrix square (A102920).
Original entry on oeis.org
1, 1, 2, 1, 4, 3, 3, 8, 9, 4, 7, 40, 27, 16, 5, 36, 152, 189, 64, 25, 6, 139, 1128, 999, 576, 125, 36, 7, 1036, 6200, 9720, 3904, 1375, 216, 49, 8, 5711, 61120, 69687, 47040, 11375, 2808, 343, 64, 9, 56355, 442552, 857466, 416704, 163500, 27432, 5145, 512
Offset: 0
Rows begin:
[1],
[1,2],
[1,4,3],
[3,8,9,4],
[7,40,27,16,5],
[36,152,189,64,25,6],
[139,1128,999,576,125,36,7],
[1036,6200,9720,3904,1375,216,49,8],
[5711,61120,69687,47040,11375,2808,343,64,9],...
The antidiagonals are formed by interleaving the
rows of triangle A102098:
[1],
[1,2],
[7,8,3],
[139,152,27,4],...
with the rows of the matrix square of A102098,
which is triangle A102920:
[1],
[3,4],
[36,40,9],
[1036,1128,189,16],...
G.f. for Column 0 (A102917): 1 = 1*(1-x) + 1*x*(1-x)
+ 1*x^2*(1-x)(1-2x) + 3*x^3*(1-x)(1-2x)
+ 7*x^4*(1-x)(1-2x)(1-3x) + 36*x^5*(1-x)(1-2x)(1-3x) +...
+ A082162(n)*x^(2n)*(1-x)(1-2x)*..*(1-(n+1)x)
+ A102921(n)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+1)x) + ...
G.f. for Column 1 (A102918): 2 = 2*(1-2x) + 4*x*(1-2x)
+ 8*x^2*(1-2x)(1-3x) + 40*x^3*(1-2x)(1-3x)
+ 152*x^4*(1-2x)(1-3x)(1-4x) + 1128*x^5*(1-2x)(1-3x)(1-4x) +...
+ T(2n+1,1)*x^(2n)*(1-2x)(1-3x)*..*(1-(n+2)x)
+ T(2n+2,1)*x^(2n+1)*(1-2x)(1-3x)*..*(1-(n+2)x) + ...
Original entry on oeis.org
1, 1, 1, 3, 7, 36, 139, 1036, 5711, 56355, 408354, 5045370, 45605881, 679409158, 7390305396, 129195427716, 1647470410551, 33114233390505, 485292763088275, 11038606786054201, 183049273155939442, 4652371578279864792
Offset: 0
1 = 1*(1-x) + 1*x*(1-x) + 1*x^2*(1-x)(1-2x) + 3*x^3*(1-x)(1-2x)
+ 7*x^4*(1-x)(1-2x)(1-3x) + 36*x^5*(1-x)(1-2x)(1-3x)
+ 139*x^6*(1-x)(1-2x)(1-3x)(1-4x) + 1036*x^7*(1-x)(1-2x)(1-3x)(1-4x) + ...
+ A082162(n)*x^(2n)*(1-x)(1-2x)*..*(1-(n+1)x)
+ A102921(n)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+1)x) + ...
-
{a(n)=if(n==0,1,polcoeff(1-sum(k=0,n-1,a(k)*x^k*prod(j=1,k\2+1,1-j*x+x*O(x^n))),n))}
Showing 1-3 of 3 results.
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