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A143122
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Triangle read by rows, T(n,k) = Sum_{j=k..n} j!, 0 <= k <= n.
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4
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1, 2, 1, 4, 3, 2, 10, 9, 8, 6, 34, 33, 32, 30, 24, 154, 153, 152, 150, 144, 120, 874, 873, 872, 870, 864, 840, 720, 5914, 5913, 5912, 5910, 5904, 5880, 5760, 5040, 46234, 46233, 46232, 46230, 46224, 46200, 46080, 45360, 40320
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OFFSET
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0,2
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COMMENTS
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Left column = A003422 starting (1, 2, 4, 10, 34, ...).
Row sums = A007489 starting (1, 3, 9, 33, 153, ...).
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LINKS
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FORMULA
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Triangle read by rows, T(n,k) = Sum_{j=k..n} j!, 0 <= k <= n.
(The above is to be read as matrix product A*D*A where A is lower triangular filled with 1's and D = diag(n!, n >= 0). - M. F. Hasler, Aug 26 2020)
T(n, k) = t(k) - t(n + 1), where t(n) = (-1)^(n + 1)*Gamma(n + 1)*Subfactorial(-(n + 1)). - Peter Luschny, Jul 11 2024
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EXAMPLE
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First few rows of the triangle are:
1;
2, 1;
4, 3, 2;
10, 9, 8, 6;
34, 33, 32, 30, 24;
...
T(4,2) = 32 = 4! + 3! + 2! = (24 + 6 + 2).
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MAPLE
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a:=proc(n, k) local j; add(factorial(j), j=k..n) end: seq(seq(a(n, k), k=0..n), n=0..8); # Muniru A Asiru, Oct 16 2018
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MATHEMATICA
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Table[Sum[j!, {j, k, n}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 15 2018 *)
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PROG
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(PARI) for(n=0, 15, for(k=0, n, print1(sum(j=k, n, j!), ", "))) \\ G. C. Greubel, Oct 15 2018
(PARI) (A143122(n, k)=sum(j=k, n, j!)); T(n)=matrix(n++, n, i, j, i>=j)*matrix(n, n, i, j, (i>=j)*i--!) \\ M. F. Hasler, Aug 26 2020
(Magma) [[(&+[Factorial(j): j in [k..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Oct 15 2018
(GAP) Flat(List([0..8], n->List([0..n], k->Sum([k..n], j->Factorial(j))))); # Muniru A Asiru, Oct 16 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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