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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

Table of n, a(n) for n=1..58.

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-Fran├žois 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.)