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) + ...
A102921
Column 0 of triangle A102920, which equals the matrix square of A102098.
Original entry on oeis.org
1, 3, 36, 1036, 56355, 5045370, 679409158, 129195427716, 33114233390505, 11038606786054201, 4652371578279864792, 2423023045813285312020, 1530233703568825263174101, 1153422053136775523883308988
Offset: 0
-
{a(n)=if(n==0,2,polcoeff(2-sum(k=0,n-1,a(k)*x^k*prod(j=2,k\2+2,1-j*x +x*O(x^n))),n))} /* at even-indices n=0,2,4,.. */
A102922
Column 1 of triangle A102920, which equals the matrix square of A102098.
Original entry on oeis.org
0, 4, 40, 1128, 61120, 5466320, 735847800, 139910204080, 35858685086352, 11953187179149408, 5037776918810353960, 2623732639426967662648, 1656984556235159516822400, 1248959074762601252295551168
Offset: 0
-
{a(n)=local(A=matrix(2,2),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,if(j==1,B[i,j]=(A^3)[i-1,1], B[i,j]=(A^3)[i-1,j]));));A=B);return((A^2)[n+1,2])}
A102923
Row sums of triangle A102920, which equals the matrix square of A102098.
Original entry on oeis.org
1, 7, 85, 2369, 127796, 11417607, 1536493698, 292107021267, 74862835208823, 24954353268384100, 10517125257007205287, 5477412008465124456814, 3459179319447147792978276, 2607366906177104506271124036
Offset: 0
-
{a(n)=local(A=matrix(2,2),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(sum(k=0,n,(A^2)[n+1,k+1]))}
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))}
Original entry on oeis.org
0, 2, 4, 8, 40, 152, 1128, 6200, 61120, 442552, 5466320, 49399320, 735847800, 8003532512, 139910204080, 1784040237288, 35858685086352, 525504809786112, 11953187179149408, 198213959637435608, 5037776918810353960
Offset: 0
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)
+ 6200*x^6*(1-2x)(1-3x)(1-4x)(1-5x) + 61120*x^7*(1-2x)(1-3x)(1-4x)(1-5x) +...
+ A102099(n+1)*x^(2n)*(1-2x)(1-3x)*..*(1-(n+2)x)
+ A102922(n+1)*x^(2n+1)*(1-x)(1-2x)*..*(1-(n+2)x) + ...
-
{a(n)=if(n==0,2,polcoeff(2-sum(k=0,n-1,a(k)*x^k*prod(j=2,k\2+2,1-j*x+x*O(x^n))),n))}
Showing 1-6 of 6 results.
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