A053154 -id:A053154 - cp's OEIS frontend

A083324 a(n) = 4^n - 3^n + 2^n.

1, 3, 11, 45, 191, 813, 3431, 14325, 59231, 242973, 990551, 4019205, 16249871, 65522733, 263668871, 1059425685, 4251986111, 17050860093, 68332318391, 273716169765, 1096025891951, 4387588255053, 17560809179111, 70274609387445, 281192563951391, 1125052651787613

Offset: 0

Paul Barry, Apr 27 2003

An alternating sum of decreasing powers.

Binomial transform of A083323.

Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A053154, A083323.

Table[4^n-3^n+2^n,{n,0,23}] (* Geoffrey Critzer, Dec 01 2013 *)

a(n) = 2 * A053154(n) + 1.
G.f.: (1-6*x+10*x^2)/((1-2*x)*(1-3*x)*(1-4*x)).
E.g.f.: exp(4*x) - exp(3*x) + exp(2*x).
a(n) = 9*a(n-1) - 26*a(n-2) + 24*a(n-3). - Geoffrey Critzer, Dec 01 2013

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

Wolfdieter Lang, Oct 27 2011

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.

			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.

Cf. A196848, A196837.

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.

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.

A052389 Number of 4-element intersecting families (with not necessarily distinct sets) of an n-element set.

Original entry on oeis.org

0, 1, 9, 95, 1286, 20681, 360964, 6452825, 114920766, 2018035121, 34864971944, 593281456505, 9965368457746, 165615181710161, 2728984827320124, 44665923097267385, 727216852411490726, 11791672548220250801

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Mar 11 2000

G. C. Greubel, Table of n, a(n) for n = 0..825
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.
Index entries for linear recurrences with constant coefficients, signature (83, -3052, 65670, -919413, 8804499, -58966886, 277278100, -904270136, 1982352768, -2749917312, 2142305280, -696729600).

Cf. A051181, A053154, A053155.

[(16^n - 6*12^n + 12*10^n - 9^n-10*8^n + 15*7^n - 24*6^n + 19*5^n + 5*4^n - 11*3^n + 6*2^n - 6)/24: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(16^n - 6*12^n + 12*10^n - 9^n-10*8^n + 15*7^n - 24*6^n + 19*5^n + 5*4^n - 11*3^n + 6*2^n - 6)/24, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((16^n - 6*12^n + 12*10^n - 9^n-10*8^n + 15*7^n - 24*6^n + 19*5^n + 5*4^n - 11*3^n + 6*2^n - 6)/24, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (16^n - 6*12^n + 12*10^n - 9^n-10*8^n + 15*7^n - 24*6^n + 19*5^n + 5*4^n - 11*3^n + 6*2^n - 6)/24.

G.f.: x * (118224000*x^10 - 215558352*x^9 + 171543508*x^8 - 77761264*x^7 + 22230235*x^6 - 4199119*x^5 + 532266*x^4 - 44801*x^3 + 2400*x^2 - 74*x + 1) / ( (x-1) * (2*x-1) * (3*x-1) * (4*x-1) * (5*x-1) * (6*x-1) * (7*x-1) * (8*x-1) * (9*x-1) * (10*x-1) * (12*x-1) * (16*x-1) ). - Colin Barker, Jul 30 2012

A083327 a(n) = (5^n - 4^n + 3^n - 2^n)/2.

Original entry on oeis.org

0, 1, 7, 40, 217, 1156, 6097, 31900, 165697, 855076, 4387537, 22404460, 113945377, 577590196, 2919923377, 14729076220, 74167952257, 372944296516, 1873182473617, 9399885079180, 47135702874337, 236224784974036

Offset: 0

Paul Barry, Apr 27 2003

Binomial transform of A053154 (with leading zero).

Index entries for linear recurrences with constant coefficients, signature (14,-71,154,-120).

Cf. A083328.

Table[(5^n-4^n+3^n-2^n)/2,{n,0,30}] (* or *) LinearRecurrence[{14,-71,154,-120},{0,1,7,40},30] (* Harvey P. Dale, Apr 04 2013 *)

G.f.: x(1-7x+13x^2)/((1-2x)(1-3x)(1-4x)(1-5x)).
E.g.f.: exp(5x) - exp(4x) + exp(3x) - exp(2x).
a(n) = 14*a(n-1) - 71*a(n-2) + 154*a(n-3) - 120*a(n-4), n > 3. - Harvey P. Dale, Apr 04 2013

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

A083324 a(n) = 4^n - 3^n + 2^n.

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

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

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A052389 Number of 4-element intersecting families (with not necessarily distinct sets) of an n-element set.

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A083327 a(n) = (5^n - 4^n + 3^n - 2^n)/2.

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

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Maple

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Formula