A117269
Triangle T, read by rows, that satisfies matrix equation: T - (T-I)^2 = C, where C is Pascal's triangle.
Original entry on oeis.org
1, 1, 1, 3, 2, 1, 19, 9, 3, 1, 207, 76, 18, 4, 1, 3211, 1035, 190, 30, 5, 1, 64383, 19266, 3105, 380, 45, 6, 1, 1581259, 450681, 67431, 7245, 665, 63, 7, 1, 45948927, 12650072, 1802724, 179816, 14490, 1064, 84, 8, 1, 1541641771, 413540343, 56925324, 5408172
Offset: 0
Triangle T begins:
1;
1,1;
3,2,1;
19,9,3,1;
207,76,18,4,1;
3211,1035,190,30,5,1;
64383,19266,3105,380,45,6,1;
1581259,450681,67431,7245,665,63,7,1; ...
where (T-I)^2 =
0;
0,0;
2,0,0;
18,6,0,0;
206,72,12,0,0;
3210,1030,180,20,0,0;
64382,19260,3090,360,30,0,0; ...
and T - (T-I)^2 = Pascal's triangle.
-
{T(n,k)=local(C=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1))),M=C); for(i=1,n+1,M=(M-M^0)^2+C);return(M[n+1,k+1])}
A117270
Matrix log of triangle M = A117269, which satisfies: M - (M-I)^2 = C where C is Pascal's triangle.
Original entry on oeis.org
0, 1, 0, 2, 2, 0, 12, 6, 3, 0, 134, 48, 12, 4, 0, 2100, 670, 120, 20, 5, 0, 42302, 12600, 2010, 240, 30, 6, 0, 1041852, 296114, 44100, 4690, 420, 42, 7, 0, 30331814, 8334816, 1184456, 117600, 9380, 672, 56, 8, 0, 1019056260, 272986326, 37506672, 3553368
Offset: 0
Triangle begins:
0;
1,0;
2,2,0;
12,6,3,0;
134,48,12,4,0;
2100,670,120,20,5,0;
42302,12600,2010,240,30,6,0;
1041852,296114,44100,4690,420,42,7,0; ...
-
{a(n)=local(C=matrix(n+1,n+1,r,c,if(r>=c,binomial(r-1,c-1))),M=C,L); for(i=1,n+1,M=(M-M^0)^2+C);L=sum(r=1,#M,-(M^0-M)^r/r);return(L[n+1,1])}
A118791
Triangle where T(n,k) = -n!*[x^k] ( x/log(1-x-x^2) )^(n+1), for n>=k>=0, read by rows.
Original entry on oeis.org
1, -1, 3, 2, -9, 19, -6, 36, -103, 207, 24, -180, 650, -1605, 3211, -120, 1080, -4710, 13860, -32191, 64383, 720, -7560, 38640, -132300, 351722, -790629, 1581259, -5040, 60480, -354480, 1386000, -4163166, 10433556, -22974463, 45948927, 40320, -544320, 3598560, -15830640, 53117064
Offset: 0
Triangle begins:
1;
-1, 3;
2,-9, 19;
-6, 36,-103, 207;
24,-180, 650,-1605, 3211;
-120, 1080,-4710, 13860,-32191, 64383;
720,-7560, 38640,-132300, 351722,-790629, 1581259;
-5040, 60480,-354480, 1386000,-4163166, 10433556,-22974463, 45948927;
which is formed from the powers of F(x) = x/log(1-x-x^2):
F(x)^1 = (-1) + 3/2*x - 11/12*x^2 + 9/8*x^3 - 641/720*x^4 +...
F(x)^2 = ( 1 - 3*x)/1! + 49/12*x^2 - 5*x^3 + 1439/240*x^4 +...
F(x)^3 = (-2 + 9*x - 19*x^2)/2! + 15*x^3 - 5161/240*x^4 +...
F(x)^4 = ( 6 - 36*x + 103*x^2 - 207*x^3)/3! + 42239/720*x^4 +...
F(x)^5 = (-24 + 180*x - 650*x^2 + 1605*x^3 - 3211*x^4)/4! +...
Showing 1-3 of 3 results.
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