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 A165313 Triangle T(n,k) = A091137(k-1) read by rows. 4
 1, 1, 2, 1, 2, 12, 1, 2, 12, 24, 1, 2, 12, 24, 720, 1, 2, 12, 24, 720, 1440, 1, 2, 12, 24, 720, 1440, 60480, 1, 2, 12, 24, 720, 1440, 60480, 120960, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 1, 2, 12 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS From a study of modified initialization formulas in Adams-Bashforth (1855-1883) multisteps method for numerical integration. On p. 36, a(i,j) comes from (j!)*a(i,j) = Integral_{u=-i-1..1} u*(u-1)*...*(u-j+1) du; see p. 32. Then, with i vertical, j horizontal, with unreduced fractions, partial array is: 0) 1, 1/2,-1/12, 1/24, -19/720, 27/1440, 1) 1, 3/2, 5/12, -1/24, 11/720, -11/1440, 2) 1, 5/2, 23/12, 9/24, -19/720, 11/1440, 3) 1, 7/2, 53/12, 55/24, 251/720, -27/1440, 4) 1, 9/2, 95/12, 161/24, 1901/720, 475/1440, 5) 1, 11/2, 149/12, 351/24, 6731/720, 4277/1440, ... . See A141417, A140825, A157982, horizontal numerators: A141047, vertical numerators: A000012, A005408, A141530, A157411. On p. 56, coefficients are s(l,q) = (1/q!)* integral_{u=-l-1..1} u*(u+1)*...*(u+q-1) du. Unreduced fractions array is: -1) 1, 1/2, 5/12, 9/24, 251/720, 475/1440, ( = A002657/A091137), 0) 2, 0, 4/12, 8/24, 232/720, 448/1440, 1) 3, -3/2, 9/12, 9/24, 243/720, 459/1440, 2) 4, -8/2, 32/12, 0/24, 224/720, 448/1440, 3) 5, -15/2, 85/12, -55/24, 475/720, 475/1440, ... (on p. 56 up to 6)). See A147998. Vertical numerators: A000027, A147998, A152064, A157371, A165281. REFERENCES P. Curtz, Integration numerique des systemes differentiels a conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, (1969). LINKS EXAMPLE 1; 1,2; 1,2,12; 1,2,12,24; 1,2,12,24,720; MATHEMATICA (* a = A091137 *) a[n_] := a[n] = Product[d, {d, Select[Divisors[n]+1, PrimeQ]}]*a[n-1]; a[0]=1; Table[Table[a[k-1], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Dec 18 2014 *) CROSSREFS Cf. A090624, A091137. Sequence in context: A266655 A209610 A324058 * A324121 A320834 A052579 Adjacent sequences:  A165310 A165311 A165312 * A165314 A165315 A165316 KEYWORD nonn,tabl AUTHOR Paul Curtz, Sep 14 2009 STATUS approved

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Last modified May 19 10:53 EDT 2019. Contains 323390 sequences. (Running on oeis4.)