A076014 -id:A076014 - cp's OEIS frontend

A244128 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k)*binomial(n,k).

0, 1, 0, 1, -2, 0, 1, -4, 9, 0, 1, -8, 27, -64, 0, 1, -16, 81, -256, 625, 0, 1, -32, 243, -1024, 3125, -7776, 0, 1, -64, 729, -4096, 15625, -46656, 117649, 0, 1, -128, 2187, -16384, 78125, -279936, 823543, -2097152, 0, 1, -256, 6561, -65536, 390625, -1679616, 5764801, -16777216, 43046721

Offset: 1

Stanislav Sykora, Jun 22 2014

T(n,k)=(-k)^(k-1)*k^(n-k) for k>0, while T(n,0)=0 by convention. The flattened triangle start with row 1, coefficient T(1,0).

Resembles A076014, but with added powers of 0, and with sign-alternating columns.

			The first rows of the triangle (starting at n=1):
0, 1,
0, 1, -2,
0, 1, -4, 9,
0, 1, -8, 27, -64,
0, 1, -16, 81, -256, 625,
0, 1, -32, 243, -1024, 3125, -7776,

Stanislav Sykora, Table of n, a(n) for rows 1..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(11), with b=1.

Cf. A076014, A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

seq(nmax,b)={my(v,n,k,irow);
v = vector((nmax+1)*(nmax+2)/2-1);
for(n=1,nmax,irow=n*(n+1)/2;v[irow]=0;
  for(k=1,n,v[irow+k]=(-1)^(k-1)*(k*b)^(n-1);););
return(v);}
a=seq(100,1);

A076015 Sum of the (n-1)-th powers of the first n integers.

Original entry on oeis.org

1, 3, 14, 100, 979, 12201, 184820, 3297456, 67731333, 1574304985, 40851766526, 1170684360924, 36720042483591, 1251308658130545, 46034015337733480, 1818399978159990976, 76762718946972480009

Offset: 1

Wolfdieter Lang, Oct 02 2002

a(n) is the number of length n sequences of [n] that start with their maximum value.

By symmetry, counts many other simple classes of endofunctions.

			The 3 sequences for n=2 are 11, 21, 22.
The 14 = 3^0 + 3^1 + 3^2 sequences starting with their maximum value for n=3 are 111, 211, 212, 221, 222, 311, 312, 313, 321, 322, 323, 331, 332, 333.

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A076014.
A diagonal of array A103438.
A subset of A000312.

a:=n->sum((n-j+1)^n,j=0..n): seq(a(n), n=0..17); # Zerinvary Lajos, Jun 05 2008

f[n_]:=Module[{s=0},Do[s+=a^n,{a,0,n+1}]; s]; Table[f[n],{n,20}] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2010 *)
Table[Sum[m^(n-1),{m,n}],{n,20}] (* Harvey P. Dale, Jan 26 2016 *)

vector(20, n, sum(m=1, n, m^(n-1))) \\ Michel Marcus, Jan 14 2015

a(n) = Sum_{m=1..n} m^(n-1), n >= 1.

Definition changed and example added by Olivier Gérard, Jan 28 2023

cp's OEIS Frontend

A244128 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k)*binomial(n,k).

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