A088305 -id:A088305 - cp's OEIS frontend

A106195 Riordan array (1/(1-2x), x(1-x)/(1-2*x)).

1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 20, 13, 5, 1, 32, 48, 38, 19, 6, 1, 64, 112, 104, 63, 26, 7, 1, 128, 256, 272, 192, 96, 34, 8, 1, 256, 576, 688, 552, 321, 138, 43, 9, 1, 512, 1280, 1696, 1520, 1002, 501, 190, 53, 10, 1, 1024, 2816, 4096, 4048, 2972, 1683, 743, 253, 64, 11

Offset: 0

Gary W. Adamson, Apr 24 2005; Paul Barry, May 21 2006

Extract antidiagonals from the product P * A, where P = the infinite lower triangular Pascal's triangle matrix; and A = the Pascal's triangle array:
  1, 1,  1,  1, ...
  1, 2,  3,  4, ...
  1, 3,  6, 10, ...
  1, 4, 10, 20, ...
  ...
Row sums are Fibonacci(2n+2). Diagonal sums are A006054(n+2). Row sums of inverse are A105523. Product of Pascal triangle A007318 and A046854.
A106195 with an appended column of ones = A055587. Alternatively, k-th column (k=0, 1, 2) is the binomial transform of bin(n, k).
T(n,k) is the number of ideals in the fence Z(2n) having k elements of rank 1. - Emanuele Munarini, Mar 22 2011
Subtriangle of the triangle given by (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012

			Triangle begins
   1;
   2,   1;
   4,   3,   1;
   8,   8,   4,  1;
  16,  20,  13,  5,  1;
  32,  48,  38, 19,  6, 1;
  64, 112, 104, 63, 26, 7, 1;
(0, 2, 0, 0, 0, ...) DELTA (1, 0, -1/2, 1/2, 0, 0, ...) begins :
  1;
  0,  1;
  0,  2,   1;
  0,  4,   3,   1;
  0,  8,   8,   4,  1;
  0, 16,  20,  13,  5,  1;
  0, 32,  48,  38, 19,  6, 1;
  0, 64, 112, 104, 63, 26, 7, 1. - _Philippe Deléham_, Mar 22 2012

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Column 0 = 1, 2, 4...; (binomial transform of 1, 1, 1...); column 1 = 1, 3, 8, 20...(binomial transform of 1, 2, 3...); column 2: 1, 4, 13, 38...= binomial transform of bin(n, 2): 1, 3, 6...

Cf. A001792, A001906, A002620, A007318, A029653, A049612, A055587, A078812, A208341.

a106195 n k = a106195_tabl !! n !! k
a106195_row n = a106195_tabl !! n
a106195_tabl = [1] : [2, 1] : f [1] [2, 1] where
   f us vs = ws : f vs ws where
     ws = zipWith (-) (zipWith (+) ([0] ++ vs) (map (* 2) vs ++ [0]))
                      ([0] ++ us ++ [0])
-- Reinhard Zumkeller, Dec 16 2013

[ (&+[Binomial(n-k, n-j)*Binomial(j, k): j in [0..n]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 15 2020

T := (n, k) -> hypergeom([-n+k, k+1],[1],-1):
seq(lprint(seq(simplify(T(n, k)), k=0..n)), n=0..7); # Peter Luschny, May 20 2015

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207605 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A106195 *)
(* Clark Kimberling, Feb 19 2012 *)
Table[Hypergeometric2F1[-n+k, k+1, 1, -1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 15 2020 *)

create_list(sum(binomial(i,k)*binomial(n-k,n-i),i,0,n),n,0,8,k,0,n); /* Emanuele Munarini, Mar 22 2011 */

from sympy import Poly, symbols
x = symbols('x')
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

from mpmath import hyp2f1, nprint
def T(n, k): return hyp2f1(k - n, k + 1, 1, -1)
for n in range(13): nprint([int(T(n, k)) for k in range(n + 1)]) # Indranil Ghosh, May 28 2017, after formula from Peter Luschny

[[sum(binomial(n-k,n-j)*binomial(j,k) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 15 2020

T(n,k) = Sum_{j=0..n} C(n-k,n-j)*C(j,k).
From Emanuele Munarini, Mar 22 2011: (Start)
T(n,k) = Sum_{i=0..n-k} C(k,i)*C(n-k,i)*2^(n-k-i).
T(n,k) = Sum_{i=0..n-k} C(k,i)*C(n-i,k)*(-1)^i*2^(n-k-i).
Recurrence: T(n+2,k+1) = 2*T(n+1,k+1)+T(n+1,k)-T(n,k). (End)
From Clark Kimberling, Feb 19 2012: (Start)
Define u(n,x) = u(n-1,x)+v(n-1,x), v(n,x) = u(n-1,x)+(x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1. Then v matches A106195 and u matches A207605. (End)
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1). - Philippe Deléham, Mar 22 2012
T(n+k,k) is the coefficient of x^n y^k in 1/(1-2x-y+xy). - Ira M. Gessel, Oct 30 2012
T(n, k) = A208341(n+1,n-k+1), k = 0..n. - Reinhard Zumkeller, Dec 16 2013
T(n, k) = hypergeometric_2F1(-n+k, k+1, 1 , -1). - Peter Luschny, May 20 2015
G.f. 1/(1-2*x+x^2*y-x*y). - R. J. Mathar, Aug 11 2015
Sum_{k=0..n} T(n, k) = Fibonacci(2*n+2) = A088305(n+1). - G. C. Greubel, Mar 15 2020

Edited by N. J. A. Sloane, Apr 09 2007, merging two sequences submitted independently by Gary W. Adamson and Paul Barry

A032287 "DIK" (bracelet, indistinct, unlabeled) transform of 1,2,3,4,...

Original entry on oeis.org

1, 3, 6, 13, 24, 51, 97, 207, 428, 946, 2088, 4831, 11209, 26717, 64058, 155725, 380400, 936575, 2314105, 5744700, 14300416, 35708268, 89359536, 224121973, 563126689, 1417378191, 3572884062, 9019324297, 22797540648

Offset: 1

Christian G. Bower

nonn

From Petros Hadjicostas, Jun 21 2019: (Start)
Under Bower's transforms, the input sequence c = (c(m): m >= 1) describes how each part of size m in a composition is colored. In a composition (ordered partition) of n >= 1, a part of size m is assumed to be colored with one of c(m) colors.
Under the DIK transform, we are dealing with "dihedral compositions" of n >= 1. These are equivalence classes of ordered partitions of n such that two such ordered partitions are equivalent if one can be obtained from the other by rotation or reflection.
If the input sequence is c = (c(m): m >= 1), denote the output sequence under the DIK transform by b = (b(n): n >= 1); i.e., b(n) = (DIK c)(n) for n >= 1. If C(x) = Sum_{m >= 1} c(m)*x^m is the g.f. of the input sequence c, then the g.f. of b = DIK c is Sum_{n >= 1} b(n)*x^n = -(1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - C(x^d)) + (1 + C(x))^2/(4 * (1 - C(x^2))) - (1/4).
For the current sequence (a(n): n >= 1), the input sequence is c(m) = m for all m >= 1. That is, we are dealing with the so-called "m-color dihedral compositions". Here, a(n) is the number of dihedral compositions of n where each part of size m may be colored with one of m colors. For the linear and cyclic versions of such m-color compositions, see Agarwal (2000), Gibson (2017), and Gibson et al. (2018).
Since C(x) = x/(1 - x)^2, we have Sum_{n >= 1} a(n) * x^n = (1/2) * Sum_{d >= 1} (phi(d)/d) * log((1 - x^d)^2 / (1 - 3*x^d + x^(2*d))) + (1/2) * x * (1 + x - 2*x^2 + x^3 + x^4)/((1 - x)^2 * (1 + x - x^2) * (1 - x - x^2)), which is the g.f. given by Andrew Howroyd in the PARI program below.
Note that -Sum_{d >= 1} (phi(d)/d) * log (1 - C(x^d)) = Sum_{d >= 1} (phi(d)/d) * log((1 - x^d)^2 / (1 - 3*x^d + x^(2*d))) is the g.f. of the "m-color cyclic compositions" that appear in Gibson (2017) and Gibson et al. (2018). See sequence A032198, which is the CIK transform of sequence (c(m): m >= 1) = (m: m >= 1).
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
A. K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.
C. G. Bower, Transforms (2).
Meghann Moriah Gibson, Combinatorics of compositions, Master of Science, Georgia Southern University, 2017.
Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of n-color compositions, Discrete Mathematics 341 (2018), 3209-3226.
Arnold Knopfmacher and Neville Robbins, Some properties of dihedral compositions, Util. Math. 92 (2013), 207-220.
Index entries for sequences related to bracelets

Cf. A000032, A000045, A001906, A005594, A032198, A088305, A308747.

DIK := proc(L::list)
    local  x,cidx,ncyc,d,gd,g,g2,n ;
    n := nops(L) ;
    g := add(op(i,L)*x^i,i=1..n) ;
    # wrap into the cycle index of the cyclic group C_n
    cidx := 0 ;
    for ncyc from 1 to n do
        for d in numtheory[divisors](ncyc) do
            gd := subs(x=x^d,g) ;
            cidx := cidx+1/ncyc*numtheory[phi](d)*gd^(ncyc/d) ;
        end do:
    end do:
    # cycle index is half of th eone for the cyclic group plus two
    # different branches or D_n with even or odd n
    cidx := cidx/2 ;
    g2 := subs(x=x^2,g) ;
    for ncyc from 1 to n do
        if type(ncyc,'odd') then
            cidx := cidx+ g*g2^((ncyc-1)/2)/2 ;
        else
            cidx := cidx+ (g^2*g2^((ncyc-2)/2) + g2^(ncyc/2))/4 ;
        end if;
    end do:
    taylor(cidx,x=0,nops(L)) ;
    gfun[seriestolist](%) ;
end proc:
A032287_list := proc(n)
        local ele ;
        ele := [seq(i,i=1..40)] ;
        DIK(ele) ;
end proc:
A032287_list(50) ; # R. J. Mathar, Feb 14 2025

seq[n_] := x(1 + x - 2 x^2 + x^3 + x^4)/((1 - x)^2 (1 - x - x^2)(1 + x - x^2)) + Sum[EulerPhi[d]/d Log[(1 - x^d)^2/(1 - 3 x^d + x^(2d)) + O[x]^(n+1)], {d, 1, n}] // CoefficientList[#, x]& // Rest // #/2&;
seq[30] (* Jean-François Alcover, Sep 17 2019, from PARI *)

seq(n)={Vec(x*(1 + x - 2*x^2 + x^3 + x^4)/((1 - x)^2*(1 - x - x^2)*(1 + x - x^2)) + sum(d=1, n, eulerphi(d)/d*log((1 - x^d)^2/(1 - 3*x^d + x^(2*d)) + O(x*x^n))))/2} \\ Andrew Howroyd, Jun 20 2018

From Petros Hadjicostas, Jun 21 2019: (Start)
a(n) = ( F(n+4) + (-1)^n * F(n-4) - 2 * (n + 1) + (1/n) * Sum_{d|n} phi(n/d) * L(2*d) )/2 for n >= 4, where F(n) = A000045(n) and L(n) = A000032(n) are the usual n-th Fibonacci and n-th Lucas numbers, respectively.
a(n) = (A032198(n) + A308747(n))/2 for n >= 1.
G.f.: (1/2) * Sum_{d >= 1} (phi(d)/d) * log((1 - x^d)^2 / (1 - 3*x^d + x^(2*d))) + (1/2) * x * (1 + x - 2*x^2 + x^3 + x^4)/((1 - x)^2 * (1 + x - x^2) * (1 - x - x^2)).
(End)

A329156 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).

Original entry on oeis.org

1, 1, 4, 10, 29, 72, 200, 510, 1364, 3546, 9348, 24400, 64090, 167562, 439200, 1149360, 3010349, 7879832, 20633304, 54014950, 141422328, 370239300, 969323000, 2537696160, 6643839400, 17393731933, 45537549048, 119218684970, 312119004990, 817137724392, 2139295489200, 5600747143950

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

nonn

Euler transform of A032198.

Alois P. Heinz, Table of n, a(n) for n = 0..2392
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 13.

Cf. A006951, A032198, A088305, A162891, A258210, A329157, A329163, A386706, A387017.

Convolution inverse of A329157.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
      add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(a(j)*b(n-j$2), j=0..n-1))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Jul 25 2025

nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Product[1/(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1 / (1 - x^k / (1 - x^k)^2).
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
G.f.: Product_{k>=1} 1 / (1 - x^k)^A032198(k).
G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A088305.
a(n) ~ phi^(2*n-1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 07 2019
a(2^k) = A002878(2^k-1) for all nonnegative integers k. Follows from Cor. 4.5 on page 11 of Kassel-Reutenauer paper. - Michael De Vlieger, Jul 28 2025

A093960 a(1) = 1, a(2) = 2, a(n+1) = na(1) + (n-1)a(2) + ... + (n-r)*a(r+1) + ... + a(n).

Original entry on oeis.org

1, 2, 4, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371, 312119004989, 817138163596

Offset: 1

Amarnath Murthy, May 22 2004

a(1) = a(2) = 1 gives A088305, i.e., Fibonacci numbers with even indices. This can be called 'fake Fibonacci sequence'. 4 = 3+1, 11 = 8+3, 29 = 21+8, 76 = 55+21, etc. a(n) = F(2n-2) + F(2n-4).

Except for the initial terms, this is the same as the bisection of the Lucas sequence (A002878). - Franklin T. Adams-Watters, Jul 17 2006

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000045, A002878, A088305.

[1,2] cat [Lucas(2*n-3): n in [3..30]]; // G. C. Greubel, Dec 30 2021

a[1]:=1: a[2]:=2: for n from 2 to 33 do a[n+1]:=sum((n-r)*a[r+1],r=0..n-1) od: seq(a[n],n=1..33); # Emeric Deutsch, Aug 01 2005
A093960List := proc(m) local A, P, n; A := [1,2]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(A), P[-1]]);
A := [op(A), P[-1]] od; A end: A093960List(30); # Peter Luschny, Mar 24 2022

Print[1]; Print[2]; Do[Print[Fibonacci[2*n - 2] + Fibonacci[2*n - 4]], {n, 3, 20}] (* Ryan Propper, Jun 19 2005 *)
LinearRecurrence[{3,-1},{1,2,4,11},30] (* Harvey P. Dale, Nov 17 2018 *)

Vec(x*(x-1)^2*(x+1)/(x^2-3*x+1) + O(x^100)) \\ Colin Barker, Mar 26 2015

[2^(2-n)*bool(n<3) + lucas_number2(2*n-3, 1, -1) for n in (1..30)] # G. C. Greubel, Dec 30 2021

a(n) = F(2*n-2) + F(2*n-4), where F(k) is k-th Fibonacci number, n > 2.
a(n) = 3*a(n-1) - a(n-2) for n>4. - Colin Barker, Mar 26 2015
G.f.: x*(1-x)^2*(1+x) / (1-3*x+x^2). - Colin Barker, Mar 26 2015
a(n) = 2^(2-n)*[n<3] + LucasL(2*n-3). - G. C. Greubel, Dec 30 2021

More terms from Ryan Propper, Jun 19 2005

More terms from Emeric Deutsch, Aug 01 2005

A273719 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal steps in the peaks (n>=2, k>=1).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 8, 3, 1, 1, 21, 9, 3, 1, 1, 55, 27, 10, 3, 1, 1, 144, 82, 33, 11, 3, 1, 1, 377, 251, 110, 39, 12, 3, 1, 1, 987, 770, 368, 139, 45, 13, 3, 1, 1, 2584, 2358, 1229, 495, 169, 51, 14, 3, 1, 1, 6765, 7191, 4085, 1755, 632, 200, 57, 15, 3, 1, 1

Offset: 2

Emeric Deutsch, Jun 01 2016

Number of entries in row n is n-1.
Sum of entries in row n = A082582(n).
T(n,1) = A088305(n-2) = F(2n-4) where F(n) are the Fibonacci numbers  A000045.
Sum(k*T(n,k), k>=0) = A273720(n).

			Row 4 is 3,1,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding drawings show that they have 3,1,1,2,1 horizontal steps in their peaks.
Triangle starts
1;
1,1;
3,1,1;
8,3,1,1;
21,9,3,1,1

Alois P. Heinz, Rows n = 2..150, flattened
A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Cf. A000045, A088305, A082582, A273720.

eq := G = z^2*s+z*(G-z^2*s/(1-z*s)+z^2*s^2/(1-z*s))+z*G+z^2*G+z*G*(G-z^2*s/(1-z*s)+z^2/(1-z)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 16 do seq(coeff(P[n], s, j), j = 1 .. n-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t, h) option remember; expand(
      `if`(n=0, (1-t)*z^h, `if`(t<0, 0, b(n-1, y+1, 1, 0))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0)*z^h)+
      `if`(y<1, 0, b(n-1, y, 0, `if`(max(h, t)>0, h+1, 0)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n-1))(b(n, 0$3)):
seq(T(n), n=2..16);  # Alois P. Heinz, Jun 06 2016

b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[If[n == 0, (1 - t)*z^h, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]*z^h] + If[y < 1, 0, b[n - 1, y, 0, If[Max[h, t] > 0, h + 1, 0]]]]]; T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 1, n - 1}]][b[n, 0, 0, 0]];  Table[T[n], {n, 2, 16}] // Flatten (* Jean-François Alcover, Nov 29 2016 after Alois P. Heinz *)

G.f.: G(s,z), where s marks number of horizontal steps in the peaks and z marks semiperimeter, satisfies the equation given in the Maple  program.
G.f.: G(w,z), where w marks number of horizontal steps in the peaks and z marks semiperimeter, satisfies eq. (7) of the Blecher et al. reference, where one has to set x = z and y = z.
The trivariate g.f. G = G(t,s,z), where t marks number of peaks, s marks number of horizontal steps in the peaks, and z marks semiperimeter, satisfies z*(1-z)*(1-s*z)*G^2-(1-3*z-s*z+z^2+3*s*z^2-s*z^3+t*s*z^3-t*s*z^4)*G + t*s*z^2*(1-z)^2 = 0.

A308723 Total number of parts in all m-cyclic compositions of n (where each part of size m can be colored with one of m colors).

Original entry on oeis.org

1, 4, 10, 26, 59, 160, 383, 1018, 2606, 6836, 17721, 46580, 121405, 318212, 832190, 2179358, 5702903, 14933264, 39088187, 102341134, 267915110, 701426484, 1836311925, 4807575700, 12586269265, 32951401540, 86267576506, 225851752438, 591286729907, 1548009602240, 4052739537911

Offset: 1

Petros Hadjicostas, Jun 19 2019

nonn

Unmarked cyclic compositions (originally studied by Sommerville (1909)) are equivalence classes of ordered partitions of n such that two such partitions are equivalent iff one can be obtained from the other by rotation.
The so-called "m-cyclic compositions" of n are cyclic compositions of n such that each part of size m can be colored with any one of m colors. (Colored ordered partitions were originally introduced by Agarwal (2000). The theory of Bower about transforms in the web link below is a generalization of this idea.)
If b(n) is the number of m-colored compositions of n, then (b(n): n >= 1) is the CIK transform of the sequence 1, 2, 3, ... and it has the g.f. -Sum_{n >= 1} (phi(s)/s) * log(1 - C(x^s)), where C(x) = x + 2*x^2 + 3*x^3 + 4*x^4 + ... = x/(1-x)^2. (See Bower's link about transforms for information about the CIK transform.) Thus, b(n) = A032198(n) for n >= 1, and its g.f. and formula were also derived in Gibson (2017) and Gibson et al. (2018).
In general, if c = (c(m): m >= 1) is the input sequence and (b_k(n): n >= 1) is the output sequence under the CIK[k] transform of c, then b_n = (CIK c)n = Sum{k = 1..n} (CIK[k] c)n = Sum{k = 1..n} b_k(n) for all n >= 1 (see Bower's web link on transforms).
The g.f. of (b_k(n): n >= 1) is Sum_{n >= 1} b_k(n)*x^k = (1/k)*Sum_{d|k} phi(d) * A(x^d)^(k/d). It follows that Sum_{n >= 1, k >= 1} b_k(n)*x^n*y^k = -Sum_{d >= 1} (phi(d)/d) * log(1 - y^d *A(x^d)) (with the understanding that b_k(n) = 0 for k > n).
For n >= 1, let d(n) = Sum_{k >= 1} k*b_k(n) = total number of parts of in all compositions of n under the CIK transform of c = (c(m): m >= 1). Thus, d(n) is the total number of parts in all cyclic compositions of n where each part of size m can be colored with c(m) colors.
To obtain the g.f. of (d(n): n >= 1) = (Sum_{k = 1..n} k*b_k(n): n >= 1), we differentiate the bivariate g.f. Sum_{n >= 1, k >= 1} b_k(n)*x^n*y^k w.r.t. y and set y = 1. We get Sum_{n >= 1} d(n)*x^n = Sum_{d >= 1} phi(d) * A(x^d)/(1 - A(x^d)).
In our case, A(x) = x/(1 - x)^2, so Sum_{n >= 1} d(n)*x^n = -Sum_{d >=1} phi(d) * x^d/(1 - 3*x^d + x^(2*d)), which is exactly the g.f. of the current sequence that was proved in Gibson (2017) and Gibson et al. (2018).

			We have a(1) = 1 because 1_1 is the only m-color cyclic composition of n = 1 and the total number of parts is 1.
We have a(2) = 4 because 2_1, 2_2, 1_1 + 1_1 are all the m-color cyclic compositions of 2 and the total number of parts is 1 + 1 + 2 = 4.
We have a(3) = 10 because 3_1, 3_2, 3_3, 1_1 + 2_1, 1_1 + 2_2, 1_1 + 1_1 + 1_1 are all the m-color cyclic compositions of n = 3 and the total number of parts is 1 + 1 + 1 + 2 + 2 + 3 = 10.
We have a(4) = 26 because 4_1, 4_2, 4_3, 4_4, 1_1 + 3_1, 1_1 + 3_2, 1_1 + 3_3, 2_1 + 2_1, 2_1 + 2_2, 2_2 + 2_2, 1_1 + 2_1 + 1_1, 1_1 + 2_2 + 1_1, 1_1 + 1_1 + 1_1 + 1_1 are all the m-color cyclic compositions of n = 4 and the total number of parts is 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 4 = 26.

A. K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math., 31(11) (2000), 1421-1427.
C. G. Bower, Transforms (2).
Meghann Moriah Gibson, Combinatorics of compositions, Master of Science, Georgia Southern University, 2017.
Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of n-color compositions, Discrete Mathematics, 341 (2018), 3209-3226.
D. M. Y. Sommerville, On certain periodic properties of cyclic compositions of numbers, Proc. London Math. Soc., S2-7(1) (1909), 263-313.

Cf. A001906, A032198, A088305.

a(n) = Sum_{s|n} phi(s)*A088305(n/s) = Sum_{s|n} phi(n/s)*Fibonacci(2*s) for n >= 1. (See Theorem 3.1 in Gibson et al. (2018).)
a(n) ~ (2/(3 - sqrt(5)))^n/sqrt(5) for large n. (See p. 3210 in Gibson et al. (2018).)
G.f.: Sum_{s >= 1} phi(s) * x^s/(1 - 3*x^s + x^(2*s)). (See Eq. (1.2) in Gibson et al. (2018).)

A382991 Number of compositions of n such that any part 1 at position k can be k different colors.

Original entry on oeis.org

1, 1, 3, 10, 40, 193, 1110, 7473, 57821, 505945, 4940354, 53248874, 627848885, 8037734930, 111017325473, 1645384681765, 26044845197881, 438499277779636, 7824114643731522, 147476551001255125, 2928074880767254238, 61078483577649288463, 1335438738400978511877

Offset: 0

John Tyler Rascoe, Apr 11 2025

			a(3) = 10 counts: (3), (2,1_a), (2,1_b), (1_a,2), (1_a,1_a,1_a), (1_a,1_a,1_b), (1_a,1_a,1_c), (1_a,1_b,1_a), (1_a,1_b,1_b), (1_a,1_b,1_c).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A008275, A011782, A088305, A238351, A240736, A382992.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, i+1)*`if`(j=1, i, 1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 23 2025

A_x(N) = {my(x='x+O('x^N)); Vec(1+ sum(i=1,N, prod(j=1,i, j*x + x^2/(1-x))))}
A_x(30)

G.f.: 1 + Sum_{i>0} Product_{j=1..i} ( j*x + x^2/(1-x) ).

A052567 E.g.f.: (1-x)^2/(1-3*x+x^2).

Original entry on oeis.org

1, 1, 6, 48, 504, 6600, 103680, 1900080, 39795840, 937681920, 24548832000, 706966444800, 22210346188800, 755916735974400, 27706219904563200, 1088037381150720000, 45576301518139392000, 2028445209752113152000, 95589693062063456256000, 4754884242802394308608000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 509

Cf. A000142, A001906, A005443, A088305.

spec := [S,{S=Sequence(Prod(Z,Sequence(Z),Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(1-x)^2/(1-3x+x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 06 2021 *)

E.g.f.: (-1+x)^2/(1-3*x+x^2).
Recurrence: {a(1)=1, a(0)=1, a(2)=6, (n^2+3*n+2)*a(n)+(-6-3*n)*a(n+1)+a(n+2)=0}
Sum(-1/5*(3*_alpha-2)*_alpha^(-1-n), _alpha=RootOf(_Z^2-3*_Z+1))*n!
a(n) = n! * Fibonacci(2*n) for n > 0. - Ilya Gutkovskiy, Jul 17 2021

A242551 Number of n-length words on infinite alphabet {1,2,...} such that the maximal runs of consecutive equal integers have lengths that are at least as great as the integer.

Original entry on oeis.org

1, 1, 2, 5, 11, 24, 53, 118, 261, 577, 1276, 2823, 6246, 13819, 30572, 67635, 149630, 331029, 732344, 1620187, 3584388, 7929844, 17543415, 38811782, 85864379, 189960150, 420254129, 929739922, 2056889538, 4550514023, 10067228909, 22272010878, 49272989918, 109008007822, 241161451563, 533528195645

Offset: 0

Geoffrey Critzer, May 17 2014

In other words, there is no restriction on the length of runs of 1's, the length of runs of 2's must be at least two, the length of runs of 3's must be at least three...

a(n) is the number of n-color integer compositions of n such that no adjacent parts are the same color. - John Tyler Rascoe, Jul 23 2024

			a(3)=5 because we have: 111, 122, 221, 222, 333.
a(4)=11 because we have:  1111, 1122, 1221, 1222, 2211, 2221, 2222, 3331, 1333, 3333, 4444.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A088305, A351638.

b:= proc(n, t) option remember; `if`(n=0, 1,
      `if`(t=0, 0, b(n-1, t)) +add(
      `if`(t=j, 0, b(n-j, j)), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 07 2015

n=nn=35;CoefficientList[Series[1/(1-Sum[v[i]/(1+v[i])/.v[i]->z^i/(1-z),{i,1,n}]),{z,0,nn}],z]

C_x(N)={my(x='x+O('x^N), h = 1/(1-sum(i=1,N, x^i/(1 - x + x^i)))); Vec(h)}
C_x(40) \\ John Tyler Rascoe, Jul 23 2024

G.f.: 1/(1 - Sum_{i>0} x^i/(1 - x + x^i)). - John Tyler Rascoe, Jul 23 2024

A308747 Number of achiral m-color cyclic compositions of n (that is, number of cyclic compositions of n with reflection symmetry where each part of size m can be colored with one of m colors).

Original entry on oeis.org

1, 3, 6, 13, 23, 44, 73, 131, 210, 365, 575, 984, 1537, 2611, 4062, 6877, 10679, 18052, 28009, 47315, 73386, 123933, 192191, 324528, 503233, 849699, 1317558, 2224621, 3449495, 5824220, 9030985, 15248099, 23643522, 39920141, 61899647, 104512392, 162055489, 273617107

Offset: 1

Petros Hadjicostas, Jun 21 2019

nonn

Cyclic compositions of a positive integer n are equivalence classes of ordered partitions of n such that two such partitions are equivalent if one can be obtained from the other by rotation. These were first studied by Sommerville (1909).
Symmetric cyclic compositions or circular palindromes or achiral cyclic compositions are those cyclic compositions that have at least one axis of symmetry. They were also studied by Sommerville (1909, pp. 301-304).
Let (c(m): m >= 1) be the input sequence and let b = (b(n): n >= 1) be the output sequence under the CPAL (circular palindrome) transform of c; that is, b(n) = (CPAC c)n for n >= 1. Hence, b(n) is the number of symmetric cyclic compositions of n where a part of size m can be colored with one of c(m) colors. If C(x) = Sum{m >= 1} c(m)*x^m is the g.f. of the input sequence c, then the g.f. of b = (CPAL c) is Sum_{n >= 1} b(n)*x^n = (1 + C(x))^2/(2 * (1 - C(x^2))) - (1/2).
For the current sequence, the input sequence is c(m) = m for m >= 1, and we are dealing with the so-called "m-color" compositions. m-color linear compositions were studied by Agarwal (2000), whereas m-color cyclic compositions were studied by Gibson (2017) and Gibson et al. (2018).
Thus, for the current sequence, a(n) is the number of symmetric (achiral) cyclic compositions of n where a part of size m may be colored with one of m colors (for each m >= 1).
The function A(x) = (exp(Pi*(x + 1)*I)*phi^(-x - 4) - exp(2*I*Pi*x)*phi^(4 - x) + exp(Pi*x*I)*phi^(x - 4) + phi^(x + 4))/sqrt(5) - 2*x, where phi is the golden ratio, shows that the sequence can be easily extended to all integers. - Peter Luschny, Aug 09 2020

			We have a(1) = 1 because we only have one symmetric cyclic composition of n = 1, namely 1_1 (and a part of size 1 can be colored with only one color).
We have a(2) = 3 because we have the following colored achiral cyclic compositions of n = 2: 2_1, 2_2, 1_1 + 1_1.
We have a(3) = 6 because we have the following colored achiral cyclic compositions of n = 3: 3_1, 3_2, 3_3, 1_1 + 2_1, 1_1 + 2_2, 1_1 + 1_1 + 1_1.
We have a(4) = 13 because we have the following colored achiral cyclic compositions of n = 4: 4_1, 4_2, 4_3, 4_4, 1_1 + 3_1, 1_1 + 3_2, 1_1 + 3_3, 2_1 + 2_1, 2_1 + 2_2, 2_2 + 2_2, 1_1 + 2_1 + 1_1, 1_1 + 2_2 + 1_1, 1_1 + 1_1 + 1_1 + 1_1.
We have a(5) = 23 because we have the following colored achiral cyclic compositions of n = 5:
(i) with one part: 5_1, 5_2, 5_3, 5_4, 5_5;
(ii) with two parts: 1_1 + 4_1, 1_1 + 4_2, 1_1 + 4_3, 1_1 + 4_4, 2_1 + 3_1, 2_1 + 3_2, 2_1 + 3_3, 2_2 + 3_1, 2_2 + 3_2, 2_2 + 3_3;
(iii) with three parts: 1_1 + 3_1 + 1_1, 1_1 + 3_2 + 1_1, 1_1 + 3_3 + 1_1, 2_1 + 1_1 + 2_1, 2_2 + 1_1 + 2_2;
(iv) with four parts: 1_1 + 1_1 + 2_1 + 1_1, 1_1 + 1_1 + 2_2 + 1_1 (here, the axis of symmetry passes through one of the 1's and through 2);
(v) with five parts: 1_1 + 1_1 + 1_1 + 1_1 + 1_1.

A. K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.
Christian G. Bower, Transforms (2).
Petros Hadjicostas, Generalized colored circular palindromic compositions, Moscow Journal of Combinatorics and Number Theory, 9(2) (2020), 173-186.
Meghann Moriah Gibson, Combinatorics of compositions, Master of Science, Georgia Southern University, 2017.
Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of n-color compositions, Discrete Mathematics 341 (2018), 3209-3226.
D. M. Y. Sommerville, On certain periodic properties of cyclic compositions of numbers, Proc. London Math. Soc. S2-7(1) (1909), 263-313.

Cf. A000045, A001906, A005594, A032198, A032287, A088305.

CPAL (circular palindrome) transform of 1, 2, 3, 4, ...
a(n) = 2*a(n - 1) + 2*a(n - 2) - 6*a(n - 3) + 2*a(n - 4) + 2*a(n - 5) - a(n - 6) for n >= 7 with a(1) = 1, a(2) = 3, a(3) = 6, a(4) = 13, a(5) = 23, and a(6) = 44.
a(n) = 3*a(n - 2) - a(n - 4) + 2*(n - 2) for n >= 5 with a(1) = 1, a(2) = 3, a(3) = 6, and a(4) = 13.
a(n) = Fib(n + 4) + (-1)^n * Fib(n - 4) - 2*n for n >= 4, where Fib(n) = A000045(n).
G.f.: x * (1 + x - 2*x^2 + x^3 + x^4)/((1 - x)^2 * (1 - x - x^2) * (1 + x - x^2)).

cp's OEIS Frontend

A106195 Riordan array (1/(1-2*x), x*(1-x)/(1-2*x)).

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A032287 "DIK" (bracelet, indistinct, unlabeled) transform of 1,2,3,4,...

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A329156 Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).

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A093960 a(1) = 1, a(2) = 2, a(n+1) = n*a(1) + (n-1)*a(2) + ... + (n-r)*a(r+1) + ... + a(n).

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A273719 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal steps in the peaks (n>=2, k>=1).

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A308723 Total number of parts in all m-cyclic compositions of n (where each part of size m can be colored with one of m colors).

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A382991 Number of compositions of n such that any part 1 at position k can be k different colors.

A106195 Riordan array (1/(1-2x), x(1-x)/(1-2*x)).

A093960 a(1) = 1, a(2) = 2, a(n+1) = na(1) + (n-1)a(2) + ... + (n-r)*a(r+1) + ... + a(n).