A077998 -id:A077998 - cp's OEIS frontend

A105477 Triangle read by rows: T(n,k) is the number of compositions of n into k parts when there are two kinds of part 2.

1, 2, 1, 1, 4, 1, 1, 6, 6, 1, 1, 6, 15, 8, 1, 1, 7, 23, 28, 10, 1, 1, 8, 30, 60, 45, 12, 1, 1, 9, 39, 98, 125, 66, 14, 1, 1, 10, 49, 144, 255, 226, 91, 16, 1, 1, 11, 60, 202, 437, 561, 371, 120, 18, 1, 1, 12, 72, 272, 685, 1128, 1092, 568, 153, 20, 1, 1, 13, 85, 355, 1015, 1995, 2555

Offset: 1

Emeric Deutsch, Apr 09 2005

Triangle T(n,k), 1 <= k <= n, given by (0, 2, -3/2, -1/6, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. Triangle T(n,k), 0 <= k <= n, is the Riordan array (1, x*(1+x-x^2)/(1-x)). - Philippe Deléham, Jan 25 2012

			T(4,2)=6 because we have (1,3),(3,1),(2,2),(2,2'),(2',2) and (2',2').
Triangle begins:
  1;
  2,  1;
  1,  4,  1;
  1,  6,  6,  1;
  1,  6, 15,  8,  1;
From _Philippe Deléham_, Jan 25 2012: (Start)
Triangle T(n,k) given by (0, 2, -3/2, -1/6, 2/3, 0, 0, 0, ...) DELTA (1,0,0,0,0,...) begins:
  1;
  0,  1;
  0,  2,  1;
  0,  1,  4,  1;
  0,  1,  6,  6,  1;
  0,  1,  6, 15,  8,  1; ... (End)

Row sums yield A077998.

Diagonals: A000012, A005843, A000384.

G:=t*z*(1+z-z^2)/(1-z-t*z-t*z^2+t*z^3): Gser:=simplify(series(G,z=0,15)): for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 13 do seq(coeff(P[n],t^k),k=1..n) od; # yields sequence in triangular form

G.f. = t*z*(1+z-z^2)/(1-z-t*z-t*z^2+t*z^3).
T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-2j-1, k-j-1). - Emeric Deutsch, Aug 06 2006
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-1), n > 1. - Philippe Deléham, Jan 25 2012

A199853 Expansion of (1-3x+x^3)/(1-2x-x^2+x^3).

Original entry on oeis.org

1, -1, -1, -3, -6, -14, -31, -70, -157, -353, -793, -1782, -4004, -8997, -20216, -45425, -102069, -229347, -515338, -1157954, -2601899, -5846414, -13136773, -29518061, -66326481, -149034250, -334876920, -752461609, -1690765888, -3799116465, -8536537209

Offset: 0

Philippe Deléham, Nov 11 2011

Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Cf. A006356, A077998, A180262.

RecurrenceTable[{a[1]==-1, a[2]== -1, a[3]== -3, a[n]== 2*a[n-1]  + a[n-2] - a[n-3]}, a, {n,30}] (* G. C. Greubel, Aug 13 2015 *)
CoefficientList[Series[(1-3x+x^3)/(1-2x-x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{2,1,-1},{1,-1,-1,-3},40] (* Harvey P. Dale, May 31 2021 *)

Vec((1-3*x+x^3)/(1-2*x-x^2+x^3) + O(x^40)) \\ Michel Marcus, Aug 13 2015

a(n) = 2*a(n-1) + a(n-2) - a(n-3) with a(0)=1, a(1)=-1, a(2)=-1, a(3)=-3.

a(n+1) = - A077998(n). - G. C. Greubel, Aug 14 2015

A052559 Expansion of e.g.f. (1-x)/(1 - 2*x - x^2 + x^3).

Original entry on oeis.org

1, 1, 6, 36, 336, 3720, 50400, 791280, 14232960, 287763840, 6466521600, 159826867200, 4309577395200, 125885452492800, 3960073877760000, 133473015067392000, 4798579092443136000, 183299247820136448000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..395
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 501

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)/(1-2*x-x^2+x^3) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 06 2019

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

With[{nn=20},CoefficientList[Series[(1-x)/(1-2x-x^2+x^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 14 2018 *)

my(x='x+O('x^30)); Vec(serlaplace( (1-x)/(1-2*x-x^2+x^3) )) \\ G. C. Greubel, May 06 2019

m = 30; T = taylor((1-x)/(1-2*x-x^2+x^3), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 06 2019

E.g.f.: (1-x)/(1 - 2*x - x^2 + x^3).
a(n) = 2*n*a(n-1) + n*(n-1)*a(n-2) - n*(n-1)*(n-2)*a(n-3), with a(0)=1, a(1)=1, a(2)=6.
a(n) = Sum((-1/7)*(-2*_alpha+_alpha^2-1)*_alpha^(-1-n), _alpha = RootOf(_Z^3-_Z^2-2*_Z+1))*n!.
a(n) = n!*A077998(n). - R. J. Mathar, Nov 27 2011

A123508 1-dimensional quasiperiodic heptagonal sequence.

Original entry on oeis.org

1, 3, 1, 3, 2, 3, 1, 3, 2, 2, 3, 1, 3, 2, 3, 1, 3, 2, 2, 3, 2, 3, 1, 3, 2, 3, 1, 3, 2, 2, 3, 1, 3, 2, 3, 1, 3, 2, 2, 3, 2, 3, 1, 3, 2, 2, 3, 1, 3, 2, 3, 1, 3, 2, 2, 3

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 01 2006

nonn

Each iterative subset can be parsed into secondary subsets relating to A077998, a sequence generated from the Heptagonal matrix M, [1, 1, 1; 1, 1, 0; 1, 0, 0]: 1, 1, 3, 6, 14, 31, 70, ...as follows (Cf. Steinbach): Performing M^n * [1,0,0] we get 3 sets of vectors, read by rows: 1, 0, 0 1, 1, 1 3, 2, 1 6, 5, 3 .. where the n-th row pertains to the n-th iterative subset of the sequence. E.g. (3, 2, 1) is the distribution of 3's, 2's and 1's in (3,1,3,2,2,3). Furthermore, the vectors generated from M relate to the Heptagon diagonals as follows: (E.g.: given the Heptagon diagonals a = 2.24697960...(1 + 2*Cos 2Pi/7); b = 1,80193773...(2*Cos Pi/7) and c = 1 (the edge); then select any 3-termed row in the vectors, such as row 4, (6, 5, 3). Then a^4 = 6*a + 5*b + 3*1.

			1=>3; 3=>1,3,2; then the previous subset generates 3,1,3,2,2,3. The resulting subsets are (1), (1,3,2), (3,1,3,2,2,3)...which we combine to form a continuous sequence.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, p. 29.

Cf. A077998.

Let a(n) = 1; then iterate using the rules 1=>3; 2=>2,3; 3=>1,3,2; Append each successive iterate to the right, creating an infinite string.

A052613 E.g.f. (1-2x)/(1-2x-x^2+x^3).

Original entry on oeis.org

1, 0, 2, 6, 72, 720, 10080, 156240, 2822400, 56972160, 1280966400, 31654022400, 853580851200, 24932991283200, 784343085926400, 26435945023488000, 950417730662400000, 36304660098330624000, 1468365202287599616000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 558

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

With[{nn=20},CoefficientList[Series[(1-2x)/(1-2x-x^2+x^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 08 2015 *)

E.g.f.: -(-1+2*x)/(x^3-x^2-2*x+1)
Recurrence: {a(1)=0, a(0)=1, a(2)=2, (n^3+6*n^2+11*n+6)*a(n) +(-n^2-5*n-6)*a(n+1) +(-2*n-6)*a(n+2) +a(n+3)=0}
Sum(1/7*(-1+3*_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^3-_Z^2-2*_Z+1))*n!
a(n)=n!*A077998(n-2), n>=2. - R. J. Mathar, Jun 03 2022

A052640 E.g.f. x(1-x)/(1-2x-x^2+x^3).

Original entry on oeis.org

0, 1, 2, 18, 144, 1680, 22320, 352800, 6330240, 128096640, 2877638400, 71131737600, 1917922406400, 56024506137600, 1762396334899200, 59401108166400000, 2135568241078272000, 81575844571533312000

Offset: 0

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 586

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

With[{nn=20},CoefficientList[Series[-x*(-1+x)/(x^3-x^2-2*x+1),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 10 2023 *)

E.g.f.: -x*(-1+x)/(x^3-x^2-2*x+1)
Recurrence: {a(1)=1, a(0)=0, a(2)=2, (n^3+6*n^2+11*n+6)*a(n)+(-n^2-5*n-6)*a(n+1)+(-2*n-6)*a(n+2)+a(n+3)=0}
Sum(1/7*(-_alpha+_alpha^2+1)*_alpha^(-1-n), _alpha=RootOf(_Z^3-_Z^2-2*_Z+1))*n!
a(n) = n!*A077998(n-1), n>0. - R. J. Mathar, Nov 27 2011

A052971 Expansion of (1-x)/(1-2x-2x^3+2*x^4).

Original entry on oeis.org

1, 1, 2, 6, 12, 26, 60, 132, 292, 652, 1448, 3216, 7152, 15896, 35328, 78528, 174544, 387952, 862304, 1916640, 4260096, 9468896, 21046464, 46779840, 103977280, 231109696, 513686144, 1141767168, 2537799168, 5640751232, 12537664512, 27867393024, 61940690176

Offset: 0

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

a(n) is the number of compositions of n using three colors of 3. Compare to A077998 which gives the number of compositions of n using two colors of 2. - Greg Dresden and Yushu Fan, Aug 15 2023

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1043
Index entries for linear recurrences with constant coefficients, signature (2,0,2,-2).

Cf. A077998.

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

CoefficientList[Series[(1-x)/(1-2x-2x^3+2x^4),{x,0,30}],x] (* or *) LinearRecurrence[{2,0,2,-2},{1,1,2,6},32] (* Harvey P. Dale, Jul 23 2012 *)

G.f.: -(-1+x)/(1-2*x-2*x^3+2*x^4).
Recurrence: {a(1)=1, a(0)=1, a(3)=6, a(2)=2, 2*a(n)-2*a(n+1)-2*a(n+3)+a(n+4)=0}.
Sum(-1/227*(-29-50*_alpha+45*_alpha^3-14*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-2*_Z-2*_Z^3+2*_Z^4)).

More terms from James Sellers, Jun 06 2000

A120771 Expansion of ( 1-x^3+x^4+x^5-x^8 ) / ( 1-2*x^3-x^6+x^9 ).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 2, 1, 6, 5, 3, 14, 11, 6, 31, 25, 14, 70, 56, 31, 157, 126, 70, 353, 283, 157, 793, 636, 353, 1782, 1429, 793, 4004, 3211, 1782, 8997, 7215, 4004, 20216, 16212, 8997, 45425, 36428, 20216, 102069, 81853, 45425, 229347, 183922, 102069, 515338, 413269, 229347, 1157954, 928607, 515338

Offset: 0

Gary W. Adamson, Jul 03 2006

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,1,0,0,-1).

Cf. A077998 (trisection), A006054 (trisection), A006356 (trisection), A038196.

CoefficientList[Series[(1-x^3+x^4+x^5-x^8)/(1-2*x^3-x^6+x^9),{x,0,60}],x] (* or *) LinearRecurrence[{0,0,2,0,0,1,0,0,-1},{1,0,0,1,1,1,3,2,1},60] (* Harvey P. Dale, Feb 19 2016 *)

Three consecutive coefficients are generated from the left row of the n-th power of the matrix [1,1,1; 1,1,0; 1,0,0].

A121302 Number of directed column-convex polyominoes having at least one 1-cell column.

Original entry on oeis.org

1, 1, 4, 10, 28, 75, 202, 540, 1440, 3828, 10153, 26875, 71021, 187421, 494013, 1300844, 3422509, 8998118, 23642479, 62088032, 162978242, 427648023, 1121766397, 2941697012, 7712415568, 20215976824, 52981414253, 138831400836

Offset: 1

Emeric Deutsch, Aug 04 2006

a(n) = Fibonacci(2n-1) - A121469(n,0) (obviously, since A121469(n,k) is the number of directed column-convex polyominoes of area n having k 1-cell columns). Column 1 of A121301.

			a(3)=4 because, with the exception of the 3-cell column, all the other four directed column-convex polyominoes of area 3 have a 1-cell column.

E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-2,4,-1).

Cf. A121469, A121301.

G:=z*(1-z)*(1-3*z+2*z^2)/(1-3*z+z^2)/(1-2*z-z^2+z^3): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=1..32);

Vec(z*(1-z)*(1-3*z+2*z^2)/((1-3*z+z^2)*(1-2*z-z^2+z^3)) + O(z^40)) \\ Michel Marcus, Feb 14 2016

G.f.: z(1-z)(1-3z+2z^2)/[(1-3z+z^2)(1-2z-z^2+z^3)].

a(n) = A001519(n)-A077998(n-2), n>0. - R. J. Mathar, Jul 22 2022

A362379 Convolution triangle of A052547(n).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 1, 4, 0, 1, 5, 2, 6, 0, 1, 5, 14, 3, 8, 0, 1, 14, 14, 27, 4, 10, 0, 1, 19, 49, 27, 44, 5, 12, 0, 1, 42, 68, 113, 44, 65, 6, 14, 0, 1, 66, 175, 159, 214, 65, 90, 7, 16, 0, 1, 131, 286, 465, 304, 360, 90, 119, 8, 18, 0, 1

Offset: 0

Philippe Deléham, Apr 20 2023

			Triangle begins, for n>=0, 0<=k<=n :
   1 ;
   0,  1 ;
   2,  0,   1 ;
   1,  4,   0,  1 ;
   5,  2,   6,  0,  1 ;
   5, 14,   3,  8,  0,  1 ;
  14, 14,  27,  4, 10,  0,  1 ;
  19, 49,  27, 44,  5, 12,  0, 1 ;
  42, 68, 113, 44, 65,  6, 14, 0, 1 ;
  ...

Cf. A052547, A077998 (row sums), A052964 (diagonal sums).

T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k) - T(n-2,k-1) - T(n-3,k) ; T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,1) = 0, T(2,0) = 2, T(n,k) = 0 if k<0 or if k>n .
Sum_{k = 0..n} T(n,k)*x^k = A052547(n), A077998(n), A052536(n), A052941(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..n} T(n,k)*2^(n-k) = A139818(n+1) = A001045(n+1)^2.

cp's OEIS Frontend

A105477 Triangle read by rows: T(n,k) is the number of compositions of n into k parts when there are two kinds of part 2.

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A199853 Expansion of (1-3*x+x^3)/(1-2*x-x^2+x^3).

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A052559 Expansion of e.g.f. (1-x)/(1 - 2*x - x^2 + x^3).

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A123508 1-dimensional quasiperiodic heptagonal sequence.

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A052613 E.g.f. (1-2x)/(1-2x-x^2+x^3).

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A052640 E.g.f. x*(1-x)/(1-2*x-x^2+x^3).

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A052971 Expansion of (1-x)/(1-2*x-2*x^3+2*x^4).

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A120771 Expansion of ( 1-x^3+x^4+x^5-x^8 ) / ( 1-2*x^3-x^6+x^9 ).

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Mathematica

A199853 Expansion of (1-3x+x^3)/(1-2x-x^2+x^3).

A052640 E.g.f. x(1-x)/(1-2x-x^2+x^3).

A052971 Expansion of (1-x)/(1-2x-2x^3+2*x^4).