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 A111594 Triangle of arctanh numbers. 4
 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 24, 0, 20, 0, 1, 0, 0, 184, 0, 40, 0, 1, 0, 720, 0, 784, 0, 70, 0, 1, 0, 0, 8448, 0, 2464, 0, 112, 0, 1, 0, 40320, 0, 52352, 0, 6384, 0, 168, 0, 1, 0, 0, 648576, 0, 229760, 0, 14448, 0, 240, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Sheffer triangle associated to Sheffer triangle A060524. For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854. The inverse matrix of A with elements a(n,m), n,m>=0, is given in A111593. In the umbral calculus notation (see the S. Roman reference) this triangle would be called associated to (1,tanh(y)). The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060524 satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0. Without the n=0 row and m=0 column and signed, this will become the Jabotinsky triangle A049218 (arctan numbers). For Jabotinsky matrices see the Knuth reference under A039692. The row polynomials p(n,x) (defined above) have e.g.f. exp(x*arctanh(y)). Exponential Riordan array [1, arctanh(x)] = [1, log(sqrt((1+x)/(1-x)))]. - Paul Barry, Apr 17 2008 Also the Bell transform of A005359. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016 LINKS Wolfdieter Lang, First 10 rows. FORMULA E.g.f. for column m>=0: ((arctanh(x))^m)/m!. a(n, m) = coefficient of x^n of ((arctanh(x))^m)/m!, n>=m>=0, else 0. a(n, m) = a(n-1, m-1) + (n-2)*(n-1)*a(n-2, m), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n `if`(n::even, n!, 0), 10); # Peter Luschny, Jan 27 2016 MATHEMATICA rows = 10; t = Table[If[EvenQ[n], n!, 0], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t]; Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *) PROG (Sage) # uses[riordan_array from A256893] riordan_array(1, atanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015 CROSSREFS Row sums: A000246. Cf. A005359, A049218, A060524, A111593. Sequence in context: A075120 A327751 A111593 * A322549 A237996 A203951 Adjacent sequences:  A111591 A111592 A111593 * A111595 A111596 A111597 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Aug 23 2005 STATUS approved

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Last modified May 26 22:58 EDT 2020. Contains 334634 sequences. (Running on oeis4.)