A117271
Column 0 of triangle A117270, which is the matrix log of triangle A117269.
Original entry on oeis.org
0, 1, 2, 12, 134, 2100, 42302, 1041852, 30331814, 1019056260, 38805685262, 1651676734092, 77703508288694, 4003868870257620, 224255353667365022, 13565588100060643932, 881405810330856632774, 61218510507062012550180
Offset: 0
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{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])}
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; ...
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{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])}
A120903
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, -1, 2, 1, 7, -3, 3, 1, -61, 28, -6, 4, 1, 751, -305, 70, -10, 5, 1, -11821, 4506, -915, 140, -15, 6, 1, 226927, -82747, 15771, -2135, 245, -21, 7, 1, -5142061, 1815416, -330988, 42056, -4270, 392, -28, 8, 1, 134341711, -46278549, 8169372, -992964, 94626, -7686, 588, -36, 9, 1, -3975839341, 1343417110
Offset: 0
Triangle T begins:
1;
1, 1;
-1, 2, 1;
7, -3, 3, 1;
-61, 28, -6, 4, 1;
751, -305, 70, -10, 5, 1;
-11821, 4506, -915, 140, -15, 6, 1;
226927, -82747, 15771, -2135, 245, -21, 7, 1;
-5142061, 1815416, -330988, 42056, -4270, 392, -28, 8, 1;
The matrix square of T less the diagonal is (T-I)^2:
0;
0, 0;
2, 0, 0;
-6, 6, 0, 0;
62, -24, 12, 0, 0;
-750, 310, -60, 20, 0, 0;
11822, -4500, 930, -120, 30, 0, 0;
where C = T + (T-I)^2 = 2*T^2 - 2*T + I.
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/* Generated by Recursion T = C - (T-I)^2 : */ {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=C-(M-M^0)^2 ); return(M[n+1, k+1])}
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/* Generated by E.G.F.: */ {T(n,k)=n!*polcoeff(polcoeff(exp(x*y)*(1 + sqrt(4*exp(x +x*O(x^n))-3))/2,n,x),k,y)}
Showing 1-3 of 3 results.
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