The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A367198 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A367198

T(n, k) = Sum_{m = 0..n-1} Stirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), where "Stirling1" are the signed Stirling numbers of the first kind.
(history; published version)

#19 by Peter Luschny at Fri Nov 10 18:01:55 EST 2023

STATUS

editing

approved

#18 by Peter Luschny at Fri Nov 10 18:01:35 EST 2023

FORMULA

T(n, n-k) = (-1)^k*Sum_{m = 0..n-1..2*} Stirling1(km+1, n-k)} *binomial(2*(k+1), m)*T(n-m, n-(m+k))*(-1)^(, m-1), for n > 3*k+2.

STATUS

approved

editing

Discussion

Fri Nov 10

18:01

Peter Luschny: Simplified.

#17 by Peter Luschny at Fri Nov 10 15:48:00 EST 2023

STATUS

proposed

approved

#16 by Peter Luschny at Fri Nov 10 14:05:39 EST 2023

STATUS

editing

proposed

Discussion

Fri Nov 10

15:18

Thomas Scheuerle: Thank you.  I will learn from you. :-)

15:21

Thomas Scheuerle: This is so much nicer now.

#15 by Peter Luschny at Fri Nov 10 14:04:53 EST 2023

FORMULA

T(n, n) = n.

T(n, 1) = A002467(n).

Sum_{k = 1..n} T(n, k) = A180191(n+1).

CROSSREFS

Cf. A002411, A002467, (first column), A000027 (main diagonal), A008275.

Cf. A180191 (n+1) (row sums), A321331 (variant with Stirling2).

Cf. A321331 (uses Stirling numbers of second kind instead).

Discussion

Fri Nov 10

14:05

Peter Luschny: Is this OK for you?

#14 by Peter Luschny at Fri Nov 10 13:57:16 EST 2023

MAPLE

T := (n, k) -> local m; add(Stirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), m = 0..n-1): seq(seq(T(n, k), k = 1..n), n = 1..9); # Peter Luschny, Nov 10 2023

Discussion

Fri Nov 10

14:00

Peter Luschny: Small standard formulas, such as the row sums or the main diagonal, are listed directly in the cross-references, not in the formula section.

#13 by Peter Luschny at Fri Nov 10 13:53:32 EST 2023

NAME

T(n, k) = Sum_{m = 0..n-1} StirlingStirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), where "StirlingStirling1" are the signed Stirling numbers of the first kind.

COMMENTS

If we want to To use the unsigned Stirling numbers we can rewrite the formula intoas: T(n, k) = Sum_{m = 0..n-1} Stirlingabs(Stirling1(m+1, k, 2))*binomial(n, m)*(-1)^(1+m+n). Replacing in this formula Stirling1 (A008275) by Stirling2 (A048993) one obtains a shifted version of A321331.

If we would replace the Stirling numbers of the first kind (A008275) here by the second kind (A048993), we will obtain A321331 instead.

STATUS

proposed

editing

Discussion

Fri Nov 10

13:54

Peter Luschny: Minor edits. 'Shifted' because A321331 has offset 0.

#12 by Michel Marcus at Fri Nov 10 09:19:19 EST 2023

STATUS

editing

proposed

#11 by Michel Marcus at Fri Nov 10 09:19:15 EST 2023

CROSSREFS

Cf. A180191 ( row sums ).

Cf. A321331 ( uses Stirling numbers of second kind instead ).

#10 by Michel Marcus at Fri Nov 10 09:18:46 EST 2023

DATA

1, 1, 2, 4, 6, 3, 15, 30, 18, 4, 76, 165, 125, 40, 5, 455, 1075, 930, 380, 75, 6, 3186, 8015, 7679, 3675, 945, 126, 7, 25487, 67536, 70042, 37688, 11550, 2044, 196, 8, 229384, 634935, 702372, 414078, 144417, 30870, 3990, 288, 9, 2293839, 6591943, 7696245, 4886390, 1885065, 463092, 73080, 7200, 405, 10

COMMENTS

If we want to use unsigned Stirling numbers we can rewrite the formula into: T(n, k) = Sum_{m = 0..n-1} Stirling(m+1, k, 2)*binomial(n, m)*(-1)^(1+m+n). If we would replace the Stirling numbers of the first kind (A008275) here by the second kind (A048993), we will obtain A321331 instead.

If we would replace the Stirling numbers of the first kind (A008275) here by the second kind (A048993), we will obtain A321331 instead.

EXAMPLE

1, .;

1, 2 .;

4, 6, 3, .;

15, 30, 18, 4, .;

76, 165, 125, 40, 5 .;

455, 1075, 930, 380, 75, 6 .;

KEYWORD

nonn,tabf,tabl,changed

STATUS

proposed

editing