The On-Line Encyclopedia of Integer Sequences (OEIS)

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A309237

Records in the indices of largest unsigned Stirling number of first kind: a(n) = smallest m such that c(m,n) = max_{k=0,1...,m} c(m,k).
(history; published version)

#10 by Peter Luschny at Thu Jul 18 14:43:18 EDT 2019

STATUS

reviewed

approved

#9 by Peter Luschny at Thu Jul 18 12:49:22 EDT 2019

STATUS

proposed

reviewed

#8 by Max Alekseyev at Wed Jul 17 22:12:31 EDT 2019

STATUS

editing

proposed

Discussion

Thu Jul 18

12:31

Peter Luschny: In c(m,n) = max_{k=0,1...,m} c(m,k) the right hand side does not depend on n?

12:44

Max Alekseyev: Yes, it does not. It's defining equality for n. Essentially it tells that the maximum is achieved at k = n.

12:45

Max Alekseyev: I meant defining identity for m. It tells that m must be such that the maximum is achieved at k = n.

#7 by Max Alekseyev at Wed Jul 17 22:12:27 EDT 2019

PROG

(PARI) { A309237(n) = my(t=prod(i=1, n-1, x+i+O(x^n)), m); m=n); while( polcoef(t, n-1)-polcoef(t, n-2) < 0, t*=x+m; m++); m; }

STATUS

proposed

editing

#6 by Max Alekseyev at Wed Jul 17 19:36:09 EDT 2019

STATUS

editing

proposed

#5 by Max Alekseyev at Wed Jul 17 19:36:02 EDT 2019

DATA

0, 1, 2, 8, 25, 73, 204, 565, 1557, 4275, 11710, 32022, 87464, 238692

COMMENTS

Ratio a(n+1)/a(n) seems to decrease and tend to exp(1) as n grows.

#4 by Max Alekseyev at Wed Jul 17 17:02:08 EDT 2019

NAME

Records in the index indices of largest unsigned Striling numbers Stirling number of first kind: a(n) = smallest m such that |stirling1c(m,n)| = max_{k=0,1...,m} |stirling1c(m,k)|.

DATA

0, 1, 2, 8, 25, 73, 204, 565, 1557, 4275, 11710, 32022, 87464

COMMENTS

Smallest m such that A065048(m-1) = |stirling1c(m,n)|.

For k in the interval [a(n),a(n+1)-1], A065048(k-1) = |stirling1c(k,n)|.

a(n+1)/a(n) seems to tend to exp(1).

EXAMPLE

n=2 is a value for index k delivering the maximum value of c(m,k) for each fixed m in the interval [a(2),a(3)-1] = [2,7]. Then, for m in [a(3),a(4)-1] = [8,24], the maximum is given by c(m,3), and so on.

PROG

(PARI) { A309237(n) = my(t=prod(i=1, n-1, x+i+O(x^n)), m); m=n; while( polcoef(t, n-1)-polcoef(t, n-2) < 0, t*=x+m; m++); m; }

#3 by Max Alekseyev at Wed Jul 17 09:15:26 EDT 2019

NAME

allocated for Max AlekseyevRecords in the index of largest unsigned Striling numbers of first kind: a(n) = smallest m such that |stirling1(m,n)| = max_{k=0,1...,m} |stirling1(m,k)|.

DATA

0, 1, 2, 8, 25, 73, 204, 565, 1557, 4275, 11710

OFFSET

0,3

COMMENTS

Smallest m such that A065048(m-1) = |stirling1(m,n)|.

For k in the interval [a(n),a(n+1)-1], A065048(k-1) = |stirling1(k,n)|.

CROSSREFS

Cf. A000254, A000399, A008275, A065048.

KEYWORD

allocated

nonn,more

AUTHOR

Max Alekseyev, Jul 17 2019

STATUS

approved

editing

#2 by Max Alekseyev at Wed Jul 17 09:15:26 EDT 2019

NAME

allocated for Max Alekseyev

KEYWORD

recycled

allocated

#1 by Russ Cox at Sun Jan 27 08:30:53 EST 2019

KEYWORD

recycled

STATUS

approved