cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A196847 Coefficient table of numerator polynomials of the ordinary generating function for the alternating power sums for the numbers 1,2,...,2*n.

Original entry on oeis.org

1, 1, -5, 7, 1, -14, 73, -168, 148, 1, -27, 298, -1719, 5473, -9162, 6396, 1, -44, 830, -8756, 56453, -227744, 562060, -778800, 468576, 1, -65, 1865, -31070, 332463, -2385305, 11612795, -37875240, 79269676, -96420480, 52148160, 1, -90, 3647, -87900, 140202
Offset: 1

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Author

Wolfdieter Lang, Oct 27 2011

Keywords

Comments

The row length sequence of this array is A005408(n-1), n >= 1: 1,3,5,7,...
This is the array for the numerator polynomials of the o.g.f. of alternating power sums of the first 2*n positive integers.
The corresponding array for the first 2*n+1 positive integers is found in A196848.
The obvious e.g.f. of a(k,2*n) := Sum_{j=1..2*n} (-1)^j * j^k is ge(n,x) := Sum_{k>=0} a(k,2*n)*(x^k)/k! = Sum_{j=1..2*n} (-1)^j * exp(j*x) = exp(x)*(exp(2*n*x) - 1)/(exp(x) + 1).
Via Laplace transformation (see the link under A196837, addendum) one finds the corresponding o.g.f.: Ge(n,x) = n*x*Pe(n,x)/Product_{j=1..2*n} (1 - j*x) with the numerator polynomial Pe(n,x) = Sum_{m=0..2*(n-1)} a(n,m)*x^m.

Examples

			n\m 0   1   2     3     4       5      6       7      8
1:  1
2:  1  -5   7
3:  1 -14  73  -168   148
4:  1 -27 298 -1719  5473   -9162   6396
5:  1 -44 830 -8756 56453 -227744 562060 -778800 468576
...
The o.g.f. for the sequence a(k,4) := -(1^k - 2^k + 3^k -4^k) = 2*A053154(k), k>=0, (n=2) is Ge(2,x) = 2*x*(1-5*x+7*x^2)/Product_{j=1..4} (1 - j*x).
a(3,2) = (S_{1,2}(4,2) + S_{3,4}(4,2) + S_{5,6}(4,2))/3 = (A196845(4,2) + A196846(4,2) + |s(5,3)|)/3 = (119+65+35)/3 = 73. Here S_{5,6}(4,2) = a_2(1,2,3,4) = |s(5,3)|, with the Stirling numbers of the first kind s(n,m) = A048994(n,m) was used.

Crossrefs

Cf. A196848, A196837.

Formula

a(n,m) = [x^m](Ge(n,x)*Product_{j=1..2*n} (1 - j*x/(n*x))), with the o.g.f. Ge(n,x) of the sequence a(k,2*n) := Sum_{j=1..2*n} (-1)^j * j^k. See a comment above.
a(n,m) = (1/n)*(-1)^m*Sum_{i=1..n} S_{2*i-1,2*i}(2*(n-1),m), n >= 1, with the (i,j)-family of number triangles S_{i,j}(n,k) defined in a comment to A196845.

A198628 Alternating sums of powers for 1,2,3,4 and 5.

Original entry on oeis.org

1, 3, 15, 81, 435, 2313, 12195, 63801, 331395, 1710153, 8775075, 44808921, 227890755, 1155180393, 5839846755, 29458152441, 148335904515, 745888593033, 3746364947235, 18799770158361, 94271405748675, 472449569948073, 2366624981836515, 11850654345690681, 59323452211439235
Offset: 0

Views

Author

Wolfdieter Lang, Oct 27 2011

Keywords

Comments

See A196848 for the e.g.f.s and o.g.f.s of such sequences for the numbers 1,2,...,2*n+1, and A196847
for the numbers 1,2,...,2*n.

Links

Crossrefs

Cf. A083323, A196847, A196848, A196837.

Programs

  • Maple
    A198628 := proc(n)
    3^n-4^n+1-2^n+5^n ;
end proc:
seq(A198628(n),n=0..20) ; # R. J. Mathar, May 11 2022
  • Mathematica
    LinearRecurrence[{15,-85,225,-274,120},{1,3,15,81,435},30] (* Harvey P. Dale, Dec 30 2024 *)

Formula

a(n) = sum(((-1)^(j+1))*j^n,j=1..5) = 1-2^n+3^n-4^n+5^n.
E.g.f.: sum(((-1)^(j+1))*exp(j*x),j=1..5) =
exp(x)*(1+exp(5*x))/(1+exp(x)).
O.g.f.: sum(((-1)^(j+1))/(1-j*x),j=1..5) =
(1-12*x+55*x^2-114*x^3+94*x^4)/product(1-j*x,j=1..5).
A formula for the numbers of the numerator polynomial is given in A196848.

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

Views

Author

Wolfdieter Lang, Oct 28 2011

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Wolfdieter Lang, Oct 28 2011

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • PARI
    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

Formula

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.
Showing 1-4 of 4 results.

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