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Revision History for A354795

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Showing entries 1-10 | older changes

A354795

Triangle read by rows. The matrix inverse of A354794. Equivalently, the Bell transform of cfact(n) = -(n - 1)! if n > 0 and otherwise 1/(-n)!.
(history; published version)

#34 by Peter Luschny at Wed Jun 15 01:14:14 EDT 2022

STATUS

proposed

approved

#33 by Werner Schulte at Tue Jun 14 16:29:01 EDT 2022

STATUS

editing

proposed

#32 by Werner Schulte at Tue Jun 14 16:28:18 EDT 2022

FORMULA

E.g.f. of column k >= 0: ((1 - t) * log(1 - t))^k / ((-1)^k * k!). - Werner Schulte, Jun 14 2022

STATUS

approved

editing

#31 by Peter Luschny at Sat Jun 11 15:11:23 EDT 2022

STATUS

editing

approved

#30 by Peter Luschny at Sat Jun 11 15:11:20 EDT 2022

FORMULA

Sum_{k=1..n} (k + x)^(k-1)*T(n, k) = binomial(n + x - 1, n-1)*(n-1)! for n >= 1. Note that for x = k this is A354796(n, k) for 0 <= k <= n and implies in particular for x = n >= 1 the identity Sum_{k=1..n} (k + n)^(k - 1)*T(n, k) = Gamma(2*n)/n! = A006963(n+1).

STATUS

approved

editing

#29 by Peter Luschny at Sat Jun 11 14:47:34 EDT 2022

STATUS

editing

approved

#28 by Peter Luschny at Sat Jun 11 14:47:29 EDT 2022

FORMULA

CROSSREFS

Cf. A354794 (matrix inverse), A176118 (row sums), A005727 (alternating row sums), A045406 (column 2), A347276 (column 3), A345651 (column 4), A298511 (central), A008296 (variant), A159333, A264428, A159075, A006963, A354796.

STATUS

approved

editing

#27 by Peter Luschny at Sat Jun 11 07:11:21 EDT 2022

STATUS

editing

approved

#26 by Peter Luschny at Sat Jun 11 07:10:47 EDT 2022

FORMULA

Sum_{k=1..n} (k + x)^(k-1)*T(n, k) = binomial(n + x - 1, n-1)*(n-1)! for n >= 1. Note that for x = k these are terms of this is A354796(n, k) for 0 <= k <= n.

#25 by Peter Luschny at Sat Jun 11 07:09:27 EDT 2022

FORMULA

Sum_{k=1..n} (k + x)^(k-1)*T(n, k) = binomial(n + x - 1, n-1)*(n-1)! for n >= 1. Note that for x = k these are terms of A354796(n, k) for 0 <= k <= n.

CROSSREFS

STATUS

approved

editing