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 A102220 Triangular matrix, read by rows, equal to [2*I - A008459]^(-1), i.e., the matrix inverse of the difference of twice the identity matrix and the triangular matrix of squared binomial coefficients. 6
 1, 1, 1, 5, 4, 1, 55, 45, 9, 1, 1077, 880, 180, 16, 1, 32951, 26925, 5500, 500, 25, 1, 1451723, 1186236, 242325, 22000, 1125, 36, 1, 87054773, 71134427, 14531391, 1319325, 67375, 2205, 49, 1, 6818444405, 5571505472, 1138150832, 103334336, 5277300, 172480, 3920, 64, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Column 0 forms A102221. Row sums form twice column 0 for n>0. Matrix logarithm is A102222. LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA T(n,k) = C(n,k)^2*A102221(n-k). T(n,0) = A102221(n). 2*A102221(n) = Sum_{k=0..n} T(n,k) for n>0. EXAMPLE Rows begin: [1], [1,1], [5,4,1], [55,45,9,1], [1077,880,180,16,1], [32951,26925,5500,500,25,1], [1451723,1186236,242325,22000,1125,36,1],... and equal the term-by-term product of column 0 with the squared binomial coefficients (A008459): [(1)1^2], [(1)1^2,(1)1^2], [(5)1^2,(1)2^2,(1)1^2], [(55)1^2,(5)3^2,(1)3^2,(1)1^2], [(1077)1^2,(55)4^2,(5)6^2,(1)4^2,(1)1^2],... The matrix inverse is [2*I - A008459]: [1], [ -1,1], [ -1,-4,1], [ -1,-9,-9,1], [ -1,-16,-36,-16,1],... MAPLE b:= proc(n) option remember; `if`(n=0, 1, add(b(n-i)*binomial(n, i)/i!, i=1..n)) end: T:= (n, k)-> binomial(n, k)^2*b(n-k)*(n-k)!: seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Sep 10 2019 MATHEMATICA nmax = 10; M = Inverse[2 IdentityMatrix[nmax+1] - Table[Binomial[n, k]^2, {n, 0, nmax}, {k, 0, nmax}]]; T[n_, k_] := M[[n+1, k+1]]; Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2019 *) PROG (PARI) {T(n, k)=(matrix(n+1, n+1, i, j, if(i==j, 2, 0)-binomial(i-1, j-1)^2)^-1)[n+1, k+1]} CROSSREFS Cf. A008459, A102221, A102222. Sequence in context: A152862 A348014 A108440 * A279151 A224228 A286680 Adjacent sequences: A102217 A102218 A102219 * A102221 A102222 A102223 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Dec 31 2004 STATUS approved

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Last modified May 23 18:34 EDT 2024. Contains 372765 sequences. (Running on oeis4.)