A196847 -id:A196847 - cp's OEIS frontend

A053154 Number of 2-element intersecting families (with not necessarily distinct sets) of an n-element set.

0, 1, 5, 22, 95, 406, 1715, 7162, 29615, 121486, 495275, 2009602, 8124935, 32761366, 131834435, 529712842, 2125993055, 8525430046, 34166159195, 136858084882, 548012945975, 2193794127526, 8780404589555, 35137304693722

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Feb 28 2000

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 11 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, 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, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A036239, A083324.

Cf. A000225, A032263, A028243.

[(4^n-3^n+2^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Oct 06 2017

Table[(4^n-3^n+2^n-1)/2, {n,1,30}] (* Clark Kimberling, Mar 12 2012 *)
CoefficientList[Series[x (1 - 5 x + 7 x^2) / ((1 - x) (1 - 4 x) (1 - 3 x) (1 - 2 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 06 2017 *)

a(n) = (4^n-3^n+2^n-1)/2; \\ Michel Marcus, Nov 30 2015

a(n) = (A083324(n) - 1)/2.
a(n) = (4^n - 3^n + 2^n - 1)/2.
a(n) = 3*StirlingS2(n+1,4) + 2*StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
From Wolfdieter Lang, Oct 28 2011 (Start)
E.g.f.: Sum_{j=1..4} ((-1)^j*exp(j*x))/2  = exp(x)*(exp(4*x)-1)/(exp(x)+1)/2.
O.g.f.: Sum_{j=1..4} (((-1)^j)/(1-j*x))/2 = x*(1-5*x+7*x^2)/product(1-j*x,j=1..4). See A196847.
(End)
G.f.: x*(1-5*x+7*x^2)/((1-x)*(1-4*x)*(1-3*x)*(1-2*x)). - Vincenzo Librandi, Oct 06 2017

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.

Original entry on oeis.org

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).

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

Wolfdieter Lang, Oct 27 2011

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.

Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Cf. A083323, A196847, A196848, A196837.

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

LinearRecurrence[{15,-85,225,-274,120},{1,3,15,81,435},30] (* Harvey P. Dale, Dec 30 2024 *)

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

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.

A376117 Irregular triangle of numerator polynomial coefficients of C({1..n},x), T(n,k) for n >= 0 and k >= A000217(n).

Original entry on oeis.org

1, 1, -2, -1, -6, 0, 10, 16, 4, -11, -17, -12, -5, -1, -24, 84, -60, 30, -144, -48, 104, 186, 268, -12, -240, -436, -348, -46, 262, 444, 391, 199, -23, -166, -207, -172, -109, -55, -21, -6, -1, 120, -1200, 4560, -7740, 5064, -2472, 9768, -19152, 35004, -39408

Offset: 0

John Tyler Rascoe, Sep 10 2024

			For row n = 2, C({1,2},x) = (-2*x^3 - x^4)/(1 + x + 2*x^2 - x^3 - x^4).
Triangle begins
  k=0  1  2  3   4   5   6   7   8   9  10   11   12   13  14  15
n=0 1;
n=1 .  1;
n=2 .  .  . -2, -1;
n=3 .  .  .  .   .   .  -6,  0, 10, 16,  4, -11, -17, -12, -5, -1;

John Tyler Rascoe, Rows n = 0..7, flattened

Cf. A107429, A175669, A196847, A231147, A373196.

C_x(s)={my( g=if(#s <1, 1, sum(i=1, #s, C_x(s[^i]) * x^(s[i]) )/(1-sum(i=1, #s, x^(s[i]))))); return(g)}
A376117_row(n)={my(t=n*(n+1)/2, c=C_x([1..n]), d=poldegree(numerator(c))-t, z=vector(d+1)); for(k=0,d,z[k+1]=polcoeff(numerator(c),k+t)); z}

C({s},x) = Sum_{i in {s}} (C({s}-{i},x)*x^i)/(1 - Sum_{i in {s}} (x^i)) with C({},x) = 1.

cp's OEIS Frontend

A053154 Number of 2-element intersecting families (with not necessarily distinct sets) of an n-element set.

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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.

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

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

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

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A376117 Irregular triangle of numerator polynomial coefficients of C({1..n},x), T(n,k) for n >= 0 and k >= A000217(n).

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