A083323 -id:A083323 - cp's OEIS frontend

A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

1, 2, 3, 10, 899

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  0  0    0        0
     {1}  {1}      {1}
          {2}{12}  {2}{12}
                   {1}{3}{23}
                   {2}{13}{23}
                   {3}{23}{123}
                   {2}{3}{13}{23}
                   {1}{3}{23}{123}
                   {2}{13}{23}{123}
                   {2}{3}{13}{23}{123}

The labeled version is A330282.
Partial sums of A330295 (the covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234.

A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 1, 7, 889

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 7 set-systems:
  0  {1}  {1}{12}  {1}{2}{13}
                   {1}{12}{23}
                   {1}{12}{123}
                   {1}{2}{12}{13}
                   {1}{2}{13}{123}
                   {1}{12}{23}{123}
                   {1}{2}{12}{13}{123}

The labeled version is A330229.
First differences of A330294 (the non-covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234, A330282.

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.

A036550 a(n) = T(0,n) + T(1,n-1) + ... + T(n,0), array T given by A048471.

Original entry on oeis.org

1, 3, 9, 29, 95, 307, 973, 3033, 9339, 28511, 86537, 261637, 788983, 2375115, 7141701, 21457841, 64439027, 193448119, 580606465, 1742343645, 5228079471, 15686335523, 47063200829, 141197991049, 423610750315, 1270865805327

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Partial sums of A083323.

[(1/2) * (3^(n+1) - 2^(n+2) + 2*n + 3): n in [0..30]]; // Vincenzo Librandi, Nov 12 2011

a(n) = (1/2) * (3^(n+1) - 2^(n+2) + 2n + 3). - Ralf Stephan, Feb 17 2004

G.f: (1/2)*(3/(1-3*x) - 4/(1-2*x) + 2*x/(1-x)^2 + 3/(1-x)). - Vincenzo Librandi, Nov 12 2011

Corrected by T. D. Noe, Nov 07 2006

A134319 Triangle read by rows. T(n, k) = binomial(n, k)*(2^k - 1 + 0^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 9, 7, 1, 4, 18, 28, 15, 1, 5, 30, 70, 75, 31, 1, 6, 45, 140, 225, 186, 63, 1, 7, 63, 245, 525, 651, 441, 127, 1, 8, 84, 392, 1050, 1736, 1764, 1016, 255, 1, 9, 108, 588, 1890, 3906, 5292, 4572, 2295, 511, 1, 10, 135, 840, 3150, 7812, 13230, 15240, 11475, 5110, 1023

Offset: 0

Gary W. Adamson, Oct 19 2007

			First few rows of the triangle:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  9,   7;
  1, 4, 18,  28,  15;
  1, 5, 30,  70,  75,  31;
  1, 6, 45, 140, 225, 186,  63;
  1, 7, 63, 245, 525, 651, 441, 127;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A083313, A083323 (row sums), A255047 (main diagonal).

x:= 'x': T:= (n,k)-> `if` (k=0, 1, abs(coeff(expand((1-1/2^x)^n -(1-2/2^x)^n), 1/(2^x)^k))): seq(seq(T(n,k), k=0..n), n=0..12); # Alois P. Heinz, Dec 10 2008
# Alternative:
T := (n, k) -> binomial(n, k)*(2^k - 1 + 0^k):
for n from 0 to 7 do seq(T(n, k), k=0..n) od;
# Or as a recursion:
p := proc(n, m) option remember; if n = 0 then max(1, m) else
    (m + x)*p(n - 1, m) - (m + 1)*p(n - 1, m + 1) fi end:
Trow := n -> seq((-1)^k * coeff(p(n, 0), x, n - k), k = 0..n):  # Peter Luschny, Jun 23 2023

max = 10; T1 = Table[Binomial[n, k], {n, 0, max}, {k, 0, max}]; T2 = Table[ If[n == k, 2^n-1, 0], {n, 0, max}, {k, 0, max}]; TT = T1.T2 ; T[, 0]=1; T[n, k_] := TT[[n+1, k+1]]; Table[T[n, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2016 *)

Previous definition: A007318 * a triangle by rows: for n > 0, n zeros followed by 2^n - 1.
Binomial transform of a diagonalized infinite lower triangular matrix with (1, 1, 3, 7, 15, ...) in the main diagonal and the rest zeros.
T(n,k) = |[1/(2^x)^k] 1 + (1-1/2^x)^n - (1-2/2^x)^n|. - Alois P. Heinz, Dec 10 2008
T(n,k) = binomial(n,k)*M(k) where M is Mersenne-like A255047. - Yuchun Ji, Feb 13 2019

More terms from Alois P. Heinz, Dec 10 2008

New name using a formula of Yuchun Ji by Peter Luschny, Jun 23 2023

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.

A122960 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 1, 0, 6, 1, 0, 0, 5, 0, 10, 1, 0, 1, 0, 15, 0, 15, 1, 0, 0, 7, 0, 35, 0, 21, 1, 0, 1, 0, 28, 0, 70, 0, 28, 1, 0, 0, 9, 0, 84, 0, 126, 0, 36, 1, 0, 1, 0, 45, 0, 210, 0, 210, 0, 45, 1

Offset: 0

Philippe Deléham, Oct 26 2006

T(n,k) = binomial (n,n-k+1) if (n-k) is an odd number (see A000217, A000332, A000579, A000581, ...). T(n,k)= 0 if (n-k)=2x with x > 0 (see A000004). T(n,n)=1 (see A000012).

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 3,  1;
  0, 1, 0,  6,  1;
  0, 0, 5,  0, 10,   1;
  0, 1, 0, 15,  0,  15,   1;
  0, 0, 7,  0, 35,   0,  21,   1;
  0, 1, 0, 28,  0,  70,   0,  28,  1;
  0, 0, 9,  0, 84,   0, 126,   0, 36,  1;
  0, 1, 0, 45,  0, 210,   0, 210,  0, 45, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000004, A000012, A000217, A000332, A000579, A000581, A007318.

T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=n then 1
    elif n=2 and k=1 then 1
    elif k=0 then 0
    else 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020

With[{m = 10}, CoefficientList[CoefficientList[Series[(1-2*x*y-x^2+x^2*y^2+
x^2*y)/(1-3*x*y-x^2+3*x^2*y^2+x^3*y-x^3*y^3), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

T(n, k) = if(k<0 || k>n, 0, if(k==n, 1, if(n==2 && k==1, 1, if(k==0, 0, 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3) )))); \\ G. C. Greubel, Feb 17 2020

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (n==2 and k==1): return 1
    elif (k==0): return 0
    else: return 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3)
print([[T(n, k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 17 2020

Sum_{k=0..n} T(n,k) = A011782(n).
Sum_{k=0..n} 2^k*T(n,k) = A083323(n).
Sum_{k=0..n} 2^(n-k)*T(n,k) = A122983(n).
G.f.: (1 - 2*x*y - x^2 + x^2*y^2 + x^2*y)/(1 - 3*x*y - x^2 + 3*x^2*y^2 + x^3*y - x^3*y^3). - Philippe Deléham, Nov 09 2013
T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 3*T(n-2,k-2) - T(n-3,k-1) + T(n-3,k-3), T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

a(12) corrected by Georg Fischer, Feb 17 2020

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

A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

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A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

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

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A036550 a(n) = T(0,n) + T(1,n-1) + ... + T(n,0), array T given by A048471.

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A134319 Triangle read by rows. T(n, k) = binomial(n, k)*(2^k - 1 + 0^k).

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

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A122960 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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

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