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| A344381
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Triangle read by rows: T(n,k) = (n+2) * (Sum_{i=k..n} i!) / ((k+2) * k!) for 0 <= k <= n with T(i,j) = 0 if j < 0 or i < j.
(history;
published version)
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#12 by N. J. A. Sloane at Thu Jun 24 02:16:56 EDT 2021
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#11 by Sean A. Irvine at Sat Jun 12 22:23:43 EDT 2021
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#10 by Sean A. Irvine at Sat Jun 12 22:23:39 EDT 2021
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| FORMULA
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Row sums are p(n; 1): {): 1, 4, 13, 46, 199, 1072, ...}., ....
Alternating row sums are p(n; -1): {): 1, 2, 5, 14, 55, 286, ...}., ....
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proposed
editing
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#9 by Amiram Eldar at Sat Jun 12 16:17:09 EDT 2021
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#8 by Amiram Eldar at Sat Jun 12 16:17:05 EDT 2021
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| MATHEMATICA
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T[n_, k_] := (n + 2) * Sum[i!, {i, k, n}]/((k + 2)*k!); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 12 2021 *)
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proposed
editing
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#7 by Werner Schulte at Mon May 17 04:00:13 EDT 2021
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#6 by Werner Schulte at Mon May 17 03:57:53 EDT 2021
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| COMMENTS
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There is a relationship to the left factorials A003422. (see formula).
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| FORMULA
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T(n,k) = (n+2) * (A003422(n+1)-A003422(k)) / ((k+2) * k!) for 0 <= k <= n.
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Discussion
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| Mon May 17
| 03:59
| Werner Schulte: Sorry, I forgot the first formula.
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#5 by Joerg Arndt at Mon May 17 02:24:36 EDT 2021
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#4 by Werner Schulte at Sun May 16 12:57:36 EDT 2021
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Discussion
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| Mon May 17
| 02:24
| Joerg Arndt: Saying "There is a relationship to the left factorials A003422." but not giving the slightest indication what it may be is not nice to the reader.
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#3 by Werner Schulte at Sun May 16 12:55:42 EDT 2021
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| COMMENTS
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There is a relationship to the left factorials A003422.
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| EXAMPLE
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The matrix inverse M (n,k), 0 <= k <= n, begins:
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| CROSSREFS
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Cf. A003422, A051403 (column 0).
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