A065048 -id:A065048 - cp's OEIS frontend

A002870 Largest Stirling numbers of second kind: a(n) = max_{k=1..n} S2(n,k).

1, 1, 3, 7, 25, 90, 350, 1701, 7770, 42525, 246730, 1379400, 9321312, 63436373, 420693273, 3281882604, 25708104786, 197462483400, 1709751003480, 15170932662679, 132511015347084, 1241963303533920, 12320068811796900, 120622574326072500, 1203163392175387500

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 835. [scanned copy]
Gábor Czédli, Four-generated direct powers of partition lattices and authentication, arXiv:2004.14509 [math.RA], 2020. See Tables 3.3 to 3.8 pp. 7-8.
Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers

Cf. A008277 (triangle of Stirling numbers of the second kind), A024417 (k at which the maximum occurs).

Cf. A065048.

a[n_] := Max[ Table[ StirlingS2[n, k], {k, 1, n}]]; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Nov 15 2011 *)

a(n) = vecmax(vector(n, k, stirling(n, k, 2))); \\ Michel Marcus, Oct 14 2015

More terms from James Sellers, Jul 10 2000

A076620 Coefficient of x^a(n) in (x+1)(x+2)...*(x+n) is the largest one.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Benoit Cloitre, Nov 10 2002

nonn

			In (x+1)(x+2)(x+3) = x^3 + 6*x^2 + 11*x + 6, the largest coefficient (11) appears at x^1, hence a(3)=1.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A065048, A130534, A309237.

a(n) = my(p=prod(j=1, n, x+j), m=vecmax(Vec(p))); for (i=0, poldegree(p), if (polcoef(p, i)==m, return(i))); \\ Michel Marcus, Feb 19 2021

first(n) = {res = vector(n); my(r = 1); v = [1]; for(i = 1, n, v = concat(0, v) + concat(v, 0)*i; for(j = r + 1, #v, if(v[j] > v[j - 1], r++ , next ); ); res[i] = r-1 ); res } \\ David A. Corneth, Feb 21 2021

from sympy import prod, Poly
from sympy.abc import x
def A076620(n):
    y = Poly(prod(x+i for i in range(1,n+1))).all_coeffs()[::-1]
    return y.index(max(y)) # Chai Wah Wu, Mar 07 2021

Is a(n) - floor(log(n)) bounded?

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).

Original entry on oeis.org

0, 1, 2, 8, 25, 73, 204, 565, 1557, 4275, 11710, 32022, 87464, 238692, 650971, 1774466

Offset: 0

Max Alekseyev, Jul 17 2019

Smallest m such that A065048(m-1) = c(m,n).
For k in the interval [a(n),a(n+1)-1], A065048(k-1) = c(k,n).
Ratio a(n+1)/a(n) seems to decrease and tend to exp(1) as n grows.

			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.

Cf. A000254, A000399, A008275, A065048.

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

a(14)-a(15) from Alexander Fritsch and Johann Peters, Dec 04 2024

A356253 a(n) is the largest coefficient of P(x) := Product_{k} (x + p_k), where (p_k) are the primes dividing n listed with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15, 32, 17, 21, 19, 24, 21, 22, 23, 44, 25, 26, 27, 32, 29, 31, 31, 80, 33, 34, 35, 60, 37, 38, 39, 68, 41, 42, 43, 48, 45, 46, 47, 112, 49, 50, 51, 56, 53, 81, 55, 92, 57, 58, 59, 92, 61, 62, 63, 240, 65, 66, 67, 72

Offset: 1

Thomas Scheuerle, Jul 31 2022

nonn

a(n) is the greatest number we may obtain by applying elementary symmetric functions onto the prime factors of n with multiplicity.
The record values of a(n)/n appear at powers of two.
If a(n) is greater than n, then it equals in most cases A003415(n), the first exception where a(n) >  A003415(n) > n is at n = 64.
Conjectured: a(A002110(n)) = A024451(n), for n > 2.
Conjecture equality breaks down after n = 175, as a(A002110(176)) > A024451(176). - Antti Karttunen, Feb 08 2024

Antti Karttunen, Table of n, a(n) for n = 1..30030

Cf. A002110, A003415, A024451, A070918, A083348, A109388, A260613, A369657 (difference between this sequence and A003415).

Cf. A065048 (same concept but uses numbers 1..n instead of prime factors of n).

a(n) = vecmax(Vec(vecprod([(x+f[1])^f[2] | f<-factor(n)~]))) \\ Edited by M. F. Hasler, Feb 14 2024

a(n) = n iff n is not in A083348, otherwise a(n) > n.
a(2^n) = A109388(n) = binomial( n, floor(n/3) )*2^(n-floor(n/3)).
a(p^n) = binomial( n, floor(n/(p+1)) )*p^(n-floor(n/(p+1))), if p is prime.
a(p*n)/a(n) >= n and <= n+1 if p is prime.
a(p*q)/a(q) = p if p and q are prime. This is also true if q is a prime greater than 7 and p is a product of two primes greater than 4.
a(A002110(n)) >= A024451(n), for n > 2. The maximum of row n in A260613 a variant of A070918.

A076622 Coefficient of x^a(n) in (x-1)(x-2)...*(x-n) is the largest one (not in absolute value).

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3

Offset: 1

Benoit Cloitre, Nov 10 2002

nonn

			(x-1)(x-2)(x-3) = x^3 - 6*x^2 + 11*x - 6, 11 is the largest coefficient for x^1, hence a(3)=1

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A065048.

N:= 200: # for a(1)..a(N)
V:= Vector(N): L:= <1>:
for n from 1 to N do
  L:= -n*  + <0, L>;
  V[n]:= max[index](L)[1]-1
od:
convert(V,list); # Robert Israel, Aug 27 2020

Is a(n)-floor(log(n)) bounded ?

A154416 Maximal Stirling numbers of the first kind.

Original entry on oeis.org

1, 1, 1, 2, 11, 35, 274, 1624, 13068, 118124, 1026576, 12753576, 120543840, 1931559552, 20313753096, 392156797824, 5056995703824, 102992244837120, 1583313975727488, 34012249593822720, 610116075740491776, 13803759753640704000, 284093315901811468800

Offset: 0

Roger L. Bagula, Jan 09 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A002870, A006551, A065048.

Table[Max[Table[StirlingS1[n, m], {m, 0, n}]], {n, 0, 30}]

a(n) = vecmax(vector(n+1, m, stirling(n, m-1, 1))); \\ Michel Marcus, Sep 16 2016

a(n) = max_{m=0..n} StirlingS1(n,m).

A369770 a(n) is the maximal coefficient in the expansion of Product_{k=1..n} (1+k*x)^k.

Original entry on oeis.org

1, 1, 8, 387, 192832, 1348952000, 142641794707200, 271057611231886800384, 10679112895658933205816311808, 9866210328276596971591655994333069312, 238373589086269734817383263830485997977600000000, 166142193793387680126634957823414405189312889036472320000000

Offset: 0

Joerg Arndt, Jan 31 2024

nonn

Cf. A065048 (maximal coefficient in Product_{k=1..n} (1+k*x) ).

b:= proc(n) b(n):= `if`(n=0, 1, expand(b(n-1)*(1+n*x)^n)) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..11);  # Alois P. Heinz, Jan 31 2024

a(n)=vecmax(Vec(prod(k=1,n,(1+k*x)^k)));
vector(20,n,a(n-1))

from collections import Counter
from math import comb
def A369770(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            a = c[j]
            for i in range(1,k+1):
                d[j+i] += comb(k,i)*k**i*a
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

A309242 a(n) is the smallest integer m such that the equality Product_{i=1..n} (x_i + m) = (m+n)!/m! for integers x_1, ..., x_n from N = { 1, 2, ..., n } guarantees that they form a permutation of N.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 9, 10, 16, 18, 27, 28, 33, 45, 45, 45, 56, 78, 81, 108, 120, 136, 140, 180, 180, 192, 209, 210, 280, 280, 286, 325, 325, 380, 392, 527, 527, 527, 625, 650, 650, 703, 703

Offset: 1

Max Alekseyev, Jul 17 2019

a(n) <= A065048(n) + 1.

JPF et al., A necessary and sufficient condition for (x_1,...,x_n) to be a permutation of (1,...,n). MathOverflow, 2019.

A349404 The maximal coefficient in the expansion of x_1(x_1 + x_2)...(x_1 + x_2 + ... + x_n).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 27, 96, 384, 1536, 7500, 37500, 194400, 1166400, 7563150, 52942050, 385351680, 3082813440, 24998984640, 224990861760, 2024917755840, 19051200000000, 190512000000000, 1944663768432000, 21391301452752000

Offset: 0

Sela Fried, Nov 16 2021

nonn

Sela Fried, On the coefficients of the distinct monomials in the expansion of x1(x1+x2)...(x1+x2+...+xn), arXiv:2111.07331 [math.CO], 2021.

Cf. A000140, A065048, A347917.

Max/@Table[Values@CoefficientRules[Times@@Array[Total@Array[x,#]&,n]],{n,0,12}] (* Giorgos Kalogeropoulos, Nov 16 2021 *)

cp's OEIS Frontend

A002870 Largest Stirling numbers of second kind: a(n) = max_{k=1..n} S2(n,k).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A076620 Coefficient of x^a(n) in (x+1)*(x+2)*...*(x+n) is the largest one.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Python

Formula

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A356253 a(n) is the largest coefficient of P(x) := Product_{k} (x + p_k), where (p_k) are the primes dividing n listed with multiplicity.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A076622 Coefficient of x^a(n) in (x-1)*(x-2)*...*(x-n) is the largest one (not in absolute value).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A154416 Maximal Stirling numbers of the first kind.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369770 a(n) is the maximal coefficient in the expansion of Product_{k=1..n} (1+k*x)^k.

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Python

A309242 a(n) is the smallest integer m such that the equality Product_{i=1..n} (x_i + m) = (m+n)!/m! for integers x_1, ..., x_n from N = { 1, 2, ..., n } guarantees that they form a permutation of N.

Views

Author

Keywords

Comments

Links

A349404 The maximal coefficient in the expansion of x_1(x_1 + x_2)...(x_1 + x_2 + ... + x_n).

A076620 Coefficient of x^a(n) in (x+1)(x+2)...*(x+n) is the largest one.

A076622 Coefficient of x^a(n) in (x-1)(x-2)...*(x-n) is the largest one (not in absolute value).