A002870 -id:A002870 - 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

A024417 s(n,a(n)) = max{s(n,k): k=1,2,...,n}, n >= 1, where s(n,k) = Stirling numbers of the second kind.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23

Offset: 1

Clark Kimberling

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A002870 (maximum values), A008277 (triangle of Stirling numbers of the second kind).

a[n_] := (m = Max[t = Table[ StirlingS2[n, k], {k, 1, n}]]; Position[t, m][[1, 1]]); Table[a[n], {n, 1, 77}] (* Jean-François Alcover, Nov 15 2011 *)

a(n) ~ n/LambertW(n) - 1 (conjecture). - Mats Granvik, Oct 16 2013

More terms retrieved from the b-file by R. J. Mathar, Sep 17 2008

A325503 Heinz number of row n of the triangle of Stirling numbers of the second kind A008277.

Original entry on oeis.org

2, 4, 20, 884, 528844, 3460086044, 340672148731996, 477782556719729075524, 11694209380474301218263758996, 4967476846044415922850025924897606724, 43298471669920632729336800855543564573041217668, 7790810575556906457316064931238939360882160372451591124244

Offset: 1

Gus Wiseman, May 07 2019

nonn

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
                              2: {1}
                              4: {1,1}
                             20: {1,1,3}
                            884: {1,1,6,7}
                         528844: {1,1,10,15,25}
                     3460086044: {1,1,15,31,65,90}
                340672148731996: {1,1,21,63,140,301,350}
          477782556719729075524: {1,1,28,127,266,966,1050,1701}
  11694209380474301218263758996: {1,1,36,255,462,2646,3025,6951,7770}

Index entries for sequences related to Heinz numbers

Cf. A000040, A001222, A002870, A008277, A024412, A056239, A112798, A215366, A325500, A325502.

Times@@@Table[Prime[StirlingS2[n,k]],{n,1,10},{k,1,n}]

a(n) = Product_{i = 1..n} prime(A008277(n,i)).
A061395(a(n)) = A002870(n).
A056239(a(n)) = A000110(n).

A024717 Sum of max{S(i,j): 1<=j<=i} for i = 1,2,...,n, where S(i,j) are Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 5, 12, 37, 127, 477, 2178, 9948, 52473, 299203, 1678603, 10999915, 74436288, 495129561, 3777012165, 29485116951, 226947600351, 1936698603831, 17107631266510, 149618646613594, 1391581950147514, 13711650761944414, 134334225088016914, 1337497617263404414

Offset: 1

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 1..576

First differences are in A002870.

A002870:= [seq(max(seq(combinat:-stirling2(i,j),j=1..i)),i=1..50)]:
ListTools:-PartialSums(A002870); # Robert Israel, Jan 23 2018

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).

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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A024417 s(n,a(n)) = max{s(n,k): k=1,2,...,n}, n >= 1, where s(n,k) = Stirling numbers of the second kind.

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A325503 Heinz number of row n of the triangle of Stirling numbers of the second kind A008277.

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Mathematica

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A024717 Sum of max{S(i,j): 1<=j<=i} for i = 1,2,...,n, where S(i,j) are Stirling numbers of the second kind.

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Author

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Links

Crossrefs

Programs

Maple

A154416 Maximal Stirling numbers of the first kind.

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Author

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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).