A198628 -id:A198628 - cp's OEIS frontend

A196848 Coefficient array of numerator polynomials of the ordinary generating functions for the alternating sums of powers for the numbers 1,2,...,2*n+1.

1, 1, -4, 5, 1, -12, 55, -114, 94, 1, -24, 238, -1248, 3661, -5736, 3828, 1, -40, 690, -6700, 40053, -151060, 351800, -465000, 270576, 1, -60, 1595, -24720, 247203, -1665900, 7660565, -23745720, 47560876, -55805520, 29400480, 1, -84, 3185, -72030, 1081353, -11344872, 85234175, -461800710, 1790256286, -4843901664, 8693117160, -9320129280, 4546558080

Offset: 0

Wolfdieter Lang, Oct 27 2011

The row length sequence of this array is A005408(n), n>=0: 1,3,5,7,...
This is the array for the numerator polynomials of the o.g.f. of alternating sums of powers of the first 2*n+1 positive integers.
The corresponding array for the first 2*n positive integers is found in A196847.
The obvious e.g.f. of a(k,2*n+1) := Sum_{j=1..2*n+1} (-1)^(j+1) * j^k is go(n,x) := Sum_{k>=0} a(k,2*n+1)*(x^k)/k! = Sum_{j=1..2*n+1} (-1)^(j+1) * exp(j*x) = exp(x)*(exp((2*n+1)*x) + 1)/(exp(x) + 1).
Via Laplace transformation (see the link under A196837, addendum) one finds the corresponding o.g.f.: Go(n,x) = Po(n,x)/Product_{j=1..2*n+1} (1 - j*x) with the numerator polynomial Po(n,x) = Sum_{m=0..2*n} a(n,m)*x^m.

			n\m 0   1   2     3     4       5      6       7       8
0:  1
1:  1  -4   5
2:  1 -12  55  -114    94
3:  1 -24 238 -1248  3661   -5736   3828
4:  1 -40 690 -6700 40053 -151060 351800 -465000, 270576
...
The o.g.f. for the sequence a(k,5) := (1^k - 2^k + 3^k - 4^k + 5^k) = A198628(k), k >= 0, (n=2) is Go(2,x) = (1 - 12*x + 55*x^2 - 114*x^3 + 94*x^4)/Product_{j=1..5} (1-j*x).
a(3,2) = S_{1,2}(5,1) + S_{3,4}(5,1) + S_{5,6}(5,1) + |s(7,5)| = A196845(5,1) + A196846(5,1) + 17 + |s(7,5)| = 25+21+17+175 = 238. Here S_{5,6}(5,1) = 1+2+3+4+7 = 17 was used.

Cf. A196847, A196837.

a(n,m) = [x^m](Go(n,x)*Product_{j=1..2*n+1} (1-j*x)), with the o.g.f. Go(n,x) of the sequence a(k,2*n+1) := Sum_{j=1..2*n+1} (-1)^(j+1) * j^k. See a comment above.

a(n,0) = 1, n >= 0, and a(n,m) = (-1)^m*((Sum_{i=1..n} S_{2*i-1,2*i}(2*(n-1),m)) + |s(2*n+1,2n+1-m)|), n >= 0, m = 1..2*n, with the (i,j)-family of number triangles S_{i,j}(n,k) defined in a comment on A196845, and the Stirling numbers of the first kind s(n,m) = A048994(n,m).

A198629 Alternating sums of powers of 1,2,...,6, divided by 3.

Original entry on oeis.org

0, 1, 7, 45, 287, 1821, 11487, 72045, 449407, 2789181, 17230367, 105996045, 649630527, 3968504541, 24174772447, 146908944045, 890924667647, 5393590283901, 32604530573727, 196853323284045, 1187295678104767, 7154833690143261

Offset: 0

Wolfdieter Lang, Oct 28 2011

For the e.g.f.s and o.g.f.s of such alternating power sums see A196847 (even case) and A196848 (odd case).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A000225, A083323, 2*A053154, A198628.

A198629 := proc(n)
    (-3^n+4^n-1+2^n-5^n+6^n)/3 ;
end proc:
seq(A198629(n),n=0..20) ; # R. J. Mathar, May 11 2022

Table[Total[Times@@@Partition[Riffle[Range[6]^n,{-1,1},{2,-1,2}],2]]/3,{n,0,30}] (* Harvey P. Dale, Jul 17 2016 *)

a(n)=sum(((-1)^j)*j^n,j=1..6)/3, n>=0.
E.g.f.: sum(((-1)^j)*exp(j*x),j=1..6)/3 = exp(x)*(exp(6*x)-1)/(3*(exp(x)+1)).
O.g.f.: sum(((-1)^j)/(1-j*x),j=1..6)/3 = x*(1-14*x+73*x^2-168*x^3+148*x^4)/
  product(1-j*x,j=1..6). See A196847 for a formula for the coefficients of the numerator polynomial.

A198630 Alternating sums of powers of 1,2,...,7.

Original entry on oeis.org

1, 4, 28, 208, 1540, 11344, 83188, 607408, 4416580, 31986064, 230784148, 1659338608, 11892395620, 84983496784, 605698755508, 4306834677808, 30560156566660

Offset: 0

Wolfdieter Lang, Oct 28 2011

For the e.g.f.s and o.g.f.s of such alternating power sums see A196847 (even case) and A196848 (odd case).

			a(2) = 1^2-2^2+3^2-4^2+5^2-6^2+7^2 = 28.

Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A000225, A083323, 2*A053154, A198628, 3*A198629.

A198630 := proc(n)
    3^n-4^n+1-2^n+5^n-6^n+7^n ;
end proc:
seq(A198630(n),n=0..20) ; # R. J. Mathar, May 11 2022

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 5040,-13068,13132,-6769,1960,-322,28]^n*[1;4;28;208;1540;11344;83188])[1,1] \\ Charles R Greathouse IV, Jul 06 2017

a(n)=sum(((-1)^(j+1))*j^n,j=1..7), n>=0.
E.g.f.: sum(((-1)^(j+1))*exp(j*x),j=1..7)= exp(x)*
  (1+exp(7*x))/(1+exp(x)).
O.g.f: sum(((-1)^(j+1))/(1-j*x),j=1..7) = (1-24*x+238*x^2-1248*x^3+3661*x^4-5736*x^5+3828*x^6)/
  product(1-j*x,j=1..7). See A196848 for a formula for the coefficients of the numerator polynomial.

cp's OEIS Frontend

A196848 Coefficient array of numerator polynomials of the ordinary generating functions for the alternating sums of powers for the numbers 1,2,...,2*n+1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A198629 Alternating sums of powers of 1,2,...,6, divided by 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A198630 Alternating sums of powers of 1,2,...,7.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula