A309237 -id:A309237 - cp's OEIS frontend

A065048 Largest unsigned Stirling number of the first kind: max_k(s(n+1,k)); i.e., largest coefficient of polynomial x(x+1)(x+2)(x+3)...*(x+n).

1, 1, 3, 11, 50, 274, 1764, 13132, 118124, 1172700, 12753576, 150917976, 1931559552, 26596717056, 392156797824, 6165817614720, 102992244837120, 1821602444624640, 34012249593822720, 668609730341153280, 13803759753640704000, 298631902863216384000

Offset: 0

Henry Bottomley, Nov 06 2001

nonn

n! <= a(n) <= (n+1)!; n <= a(n+1)/a(n) <= (n+1). - Max Alekseyev, Jul 17 2019

			a(4)=50 since polynomial is x^4 + 10*x^3 + 35*x^2 + 50*x + 24.

Robert Israel, Table of n, a(n) for n = 0..448

Cf. A000254, A000399, A002870, A008275, A309237.

P:= x: A[0]:= 1:
for n from 1 to 50 do
  P:= expand(P*(x+n));
  A[n]:= max(coeffs(P,x));
od:
seq(A[i],i=0..50); # Robert Israel, Jul 04 2016

a[n_] := Max[Array[Abs[StirlingS1[n+1, #]]&, n+1]];
Array[a, 100, 0] (* Griffin N. Macris, Jul 03 2016 *)

a(n) = if (n==0, 1, vecmax(vector(n, k, abs(stirling(n+1, k, 1))))); \\ Michel Marcus, Jul 04 2016; corrected Jun 12 2022

from collections import Counter
def A065048(n):
    c = {1:1}
    for k in range(1,n+1):
        d = Counter()
        for j in c:
            d[j] += k*c[j]
            d[j+1] += c[j]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

For n in the interval [A309237(k)-1, A309237(k+1)-2], a(n) = |Stirling1(n+1,k)|. - Max Alekseyev, Jul 17 2019

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?

A341803 a(n) is the least k such that A076620(k) = n.

Original entry on oeis.org

1, 2, 7, 24, 72, 203, 564, 1556, 4274, 11709, 32021, 87463, 238691

Offset: 0

David A. Corneth, Feb 20 2021

Conjecture: Starting at 7, terms coincide with A309237 - 1. - Hugo Pfoertner, Feb 21 2021

			a(4) = 24 as A076620(24) = 4 while A076620(k) < 4 for k < 24.

Cf. A076620.

With[{s = Array[-1 + FirstPosition[#, Max[#]][[1]] &@ CoefficientList[Pochhammer[x, #]/x, x] &, 600]}, {1}~Join~Array[-1 + FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Feb 22 2021 *)

first(n) = {res = vector(n); res[1] = 1; my(r = 1); print1(1", "); v = [1]; for(i = 1, oo, v = concat(0, v) + concat(v, 0)*i; if(#v > n, v = v[^-1]; ); for(j = r + 1, #v, if(v[j] > v[j - 1], r++; res[r] = i; print1(i", "); if(r >= n, return(res); ) , next ))); res }

a(n) ~ c*e^n.

cp's OEIS Frontend

A065048 Largest unsigned Stirling number of the first kind: max_k(s(n+1,k)); i.e., largest coefficient of polynomial x(x+1)(x+2)(x+3)...*(x+n).

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

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A341803 a(n) is the least k such that A076620(k) = n.

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cp's OEIS Frontend

A065048 Largest unsigned Stirling number of the first kind: max_k(s(n+1,k)); i.e., largest coefficient of polynomial x*(x+1)*(x+2)*(x+3)*...*(x+n).

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Author

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Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

A341803 a(n) is the least k such that A076620(k) = n.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI

Formula

A065048 Largest unsigned Stirling number of the first kind: max_k(s(n+1,k)); i.e., largest coefficient of polynomial x(x+1)(x+2)(x+3)...*(x+n).

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