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 A120903 Triangle T, read by rows, that satisfies matrix equation: T + (T-I)^2 = C, where C is Pascal's triangle. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Column 0 is signed A048287, which is the number of semiorders on n labeled nodes. LINKS FORMULA E.g.f. A(x,y) satisfies: A(x,y) + [A(x,y) - exp(x*y)]^2 = exp(x+x*y). Explicitly, e.g.f.: A(x,y) = exp(x*y)*(1 + sqrt(4*exp(x)-3))/2. E.g.f. of column 0: (1 + sqrt(4*exp(x)-3))/2. T(n,k) = -(-1)^(n-k)*A048287(n-k)*C(n,k) + 2*0^(n-k). Matrix square: [T^2](n,k) = ( C(n,k) + 2*T(n,k) - 0^(n-k) )/2. EXAMPLE 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. PROG (PARI) /* 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])} (PARI) /* 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)} CROSSREFS Cf. A048287 (column 0); A117269 (variant: T - (T-I)^2 = C). Sequence in context: A125697 A090699 A214550 * A180335 A257699 A160535 Adjacent sequences:  A120900 A120901 A120902 * A120904 A120905 A120906 KEYWORD sign,tabl AUTHOR Paul D. Hanna, Jul 17 2006 STATUS approved

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Last modified September 19 19:41 EDT 2021. Contains 347564 sequences. (Running on oeis4.)