The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A118441 Triangle L, read by rows, equal to the matrix log of A118435, with the property that L^2 consists of a single diagonal (two rows down from the main diagonal). 5
 0, 1, 0, -4, 2, 0, -12, 12, 3, 0, 32, -48, -24, 4, 0, 80, -160, -120, 40, 5, 0, -192, 480, 480, -240, -60, 6, 0, -448, 1344, 1680, -1120, -420, 84, 7, 0, 1024, -3584, -5376, 4480, 2240, -672, -112, 8, 0, 2304, -9216, -16128, 16128, 10080, -4032, -1008, 144, 9, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS L = log(A118435) = log(H*[C^-1]*H], where C=Pascal's triangle and H=A118433 where H^2 = I (identity matrix). LINKS FORMULA For even exponents of L, L^(2m) is a single diagonal: if n == k+2m, then [L^(2m)](n,k) = n!/k!*2^(n-k-2m)/(n-k-2m)!; else if n != k+2m: [L^(2m)](n,k) = 0. For odd exponents of L: if n >= k+2m+1, then [L^(2m+1)](n,k) = n!/k!*2^(n-k-2m-1)/(n-k-2m-1)!*(-1)^(m+[(n+1)/2]-[k/2]+n-k); else if n < k+2m+1: [L^(2m)](n,k) = 0. Unsigned row sums equals A027471(n+1) = n*3^(n-1). EXAMPLE The matrix log, L = log(H*[C^-1]*H], begins:      0;      1,     0;     -4,     2,      0;    -12,    12,      3,     0;     32,   -48,    -24,     4,     0;     80,  -160,   -120,    40,     5,     0;   -192,   480,    480,  -240,   -60,     6,     0;   -448,  1344,   1680, -1120,  -420,    84,     7,   0;   1024, -3584,  -5376,  4480,  2240,  -672,  -112,   8,  0;   2304, -9216, -16128, 16128, 10080, -4032, -1008, 144,  9,  0;   ... The matrix square, L^2, is a single diagonal:   0;   0, 0;   2, 0,  0;   0, 6,  0,  0;   0, 0, 12,  0,  0;   0, 0,  0, 20,  0,  0;   0, 0,  0,  0, 30,  0,  0;   ... From Peter Luschny, Apr 23 2020: (Start) In unsigned form and without the main diagonal, as computed by the Maple script:   [0], [0]   [1], [1]   [2], [4,   2]   [3], [12,  12,   3]   [4], [32,  48,   24,   4]   [5], [80,  160,  120,  40,   5]   [6], [192, 480,  480,  240,  60,  6]   [7], [448, 1344, 1680, 1120, 420, 84, 7] (End) MAPLE # Generalized Worpitzky transform of the harmonic numbers. CL := p -> PolynomialTools:-CoefficientList(expand(p), x): H := n -> add(1/k, k=1..n): Trow := proc(n) local k, v; if n=0 then return [0] fi; add(add((-1)^(n-v)*binomial(k, v)*H(k)*(-x+v-1)^n, v=0..k), k=0..n); CL(%) end: for n from 0 to 7 do Trow(n) od; # Peter Luschny, Apr 23 2020 PROG (PARI) /* From definition of L as matrix log of H*C^-1*H: */ {L(n, k)=local(H=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1)*(-1)^(r\2-(c-1)\2+r-c))), C=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1))), N=(H*C^-1*H)); Log=sum(p=1, n+1, -(N^0-N)^p/p); Log[n+1, k+1]} for(n=0, 10, for(k=0, n, print1(L(n, k), ", ")); print("")) (PARI) /* The matrix power L^m is given by: */ {L(n, k, m)=if(m%2==0, if(n==k+m, n!/k!*2^(n-k-m)/(n-k-m)!), if(n>=k+m, n!/k!*2^(n-k-m)/(n-k-m)!*(-1)^(m\2+(n+1)\2-k\2+n-k)))} for(n=0, 10, for(k=0, n, print1(L(n, k, 1), ", ")); print("")) CROSSREFS Cf. A118435 (exp(L)), A118442 (column 0), A118443 (row sums), A027471 (unsigned row sums); A118433 (self-inverse triangle), A001815 (column 1?), A001789 (third of column 2?). Sequence in context: A299769 A091435 A330472 * A244131 A206428 A334778 Adjacent sequences:  A118438 A118439 A118440 * A118442 A118443 A118444 KEYWORD sign,tabl AUTHOR Paul D. Hanna, Apr 28 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)