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A108895
Partial sums of quadruple factorial numbers n!!!! (A007662).
0
1, 2, 4, 7, 11, 16, 28, 49, 81, 126, 246, 477, 861, 1446, 3126, 6591, 12735, 22680, 52920, 118755, 241635, 450480, 1115760, 2629965, 5579085, 10800210, 28097490, 68981025, 151556385, 302969010, 821887410, 2089276995, 4731688515
OFFSET
0,2
COMMENTS
Quadruple factorial numbers n!!!! = n*(n-4)!!!!, 0!!!! = 1!!!! = 1, 2!!!! = 2, 3!!!! = 3. The cumulative sum a(n) is prime for n = 1, 3, 4 and never again, as all values from a(8) = 81 are multiples of 3. The cumulative sum a(n) is semiprime for n = 2, 7 and never again, as all values from a(16) are divisible by both 3 and 5.
REFERENCES
J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.
FORMULA
a(n) = Sum_{i=0..n} i!!!!.
a(n) = Sum_{i=0..n} A007662(i).
EXAMPLE
a(31) = 1 + 1 + 2 + 3 + 4 + 5 + 12 + 21 + 32 + 45 + 120 + 231 + 384 + 585 + 1680 + 3465 + 6144 + 9945 + 30240 + 65835 + 122880 + 208845 + 665280 + 1514205 + 2949120 + 5221125 + 17297280 + 40883535 + 82575360 + 151412625 + 518918400 + 1267389585 = 2089276995 = 3 * 5 * 13 * 337 * 31793.
MATHEMATICA
NFactorialM[n_Integer, m_Integer] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Table[ Sum[ NFactorialM[i, 4], {i, 0, n}], {n, 0, 33}] (* Robert G. Wilson v, Feb 21 2006 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 08 2006
STATUS
approved