A225685 -id:A225685

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A123892

Expansion of g.f.: (1+x^2)*(1+2*x^2)/(1-3*x+3*x^2-6*x^3+2*x^4-2*x^5).

+10
4

1, 3, 9, 24, 63, 167, 444, 1179, 3129, 8306, 22051, 58539, 155400, 412535, 1095149, 2907266, 7717839, 20488343, 54389880, 144387411, 383301505, 1017540554, 2701238539, 7170907923, 19036423288, 50535499231, 134155279397, 356138541458, 945431750839, 2509813152639

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) = number of words of length n over {0,1,2} which do not contain a factor jkj with j>k. - N. J. A. Sloane, May 21 2013

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001; Annals. Combin., 7 (2003), 1-14.

Index entries for linear recurrences with constant coefficients, signature (3,-3,6,-2,2).

FORMULA

G.f. can be written 1/(1-x*(1+1/(1+x^2)+1/(1+2*x^2))). - N. J. A. Sloane, May 21 2013

MAPLE

seq(coeff(series((1+x^2)*(1+2*x^2)/(1-3*x+3*x^2-6*x^3+2*x^4-2*x^5), x, n+1), x, n), n = 0..30); # G. C. Greubel, Aug 06 2019

MATHEMATICA

LinearRecurrence[{3, -3, 6, -2, 2}, {1, 3, 9, 24, 63}, 30] (* Jean-François Alcover, Jan 09 2019 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((1+x^2)*(1+2*x^2)/(1-3*x+3*x^2-6*x^3+2*x^4 -2*x^5)) \\ G. C. Greubel, Aug 06 2019

(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x^2)*(1+2*x^2)/(1-3*x+3*x^2-6*x^3+2*x^4-2*x^5)) )); // G. C. Greubel, Aug 06 2019

(Sage) ((1+x^2)*(1+2*x^2)/(1-3*x+3*x^2-6*x^3+2*x^4-2*x^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 06 2019

(GAP) a:=[1, 3, 9, 24, 63];; for n in [6..30] do a[n]:=3*a[n-1]-3*a[n-2] +6*a[n-3]-2*a[n-4]+2*a[n-5]; od; a; # G. C. Greubel, Aug 06 2019

CROSSREFS

Cf. A005251, A123893, A123894, A225685.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 20 2006

STATUS

approved

A123893

Expansion of g.f.: (1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2-18*x^3 +11*x^4-22*x^5+6*x^6-6*x^7).

+10
4

1, 4, 16, 58, 208, 750, 2708, 9772, 35256, 127210, 459012, 1656228, 5976040, 21562946, 77804232, 280736004, 1012961416, 3655002994, 13188110940, 47585806908, 171700784680, 619536821778, 2235434596432, 8065973894524, 29103931264328, 105013830473538

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Number of words of length n over {0,1,2,3} which have no factor iji with i>j. - N. J. A. Sloane, May 21 2013

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14.

Index entries for linear recurrences with constant coefficients, signature (4,-6,18,-11,22,-6,6).

FORMULA

a(0)=1, a(1)=4, a(2)=16, a(3)=58, a(4)=208, a(5)=750, a(6)=2708, a(n)= 4*a(n-1) -6*a(n-2) +18*a(n-3) -11*a(n-4) +22*a(n-5) -6*a(n-6) +6*a(n-7). - Harvey P. Dale, May 20 2012

G.f. can be written 1/(1-x*(1+1/(1+x^2)+1/(1+2*x^2)+1/(1+3*x^2))) which looks more symmetrical. N. J. A. Sloane, May 21 2013

MAPLE

seq(coeff(series((1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2-18*x^3+11*x^4 -22*x^5+6*x^6-6*x^7), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 06 2019

MATHEMATICA

CoefficientList[Series[-(1+x^2) (1+2 x^2) (1+3 x^2)/(-1-6 x^2-11 x^4-6 x^6+4 x+18 x^3+22 x^5+6 x^7), {x, 0, 40}], x] (* or *) LinearRecurrence[ {4, -6, 18, -11, 22, -6, 6}, {1, 4, 16, 58, 208, 750, 2708}, 40] (* Harvey P. Dale, May 20 2012 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2 -18*x^3+11*x^4-22*x^5+6*x^6-6*x^7)) \\ G. C. Greubel, Aug 06 2019

(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2-18*x^3+11*x^4-22*x^5+6*x^6-6*x^7) )); // G. C. Greubel, Aug 06 2019

(Sage) ((1+x^2)*(1+2*x^2)*(1+3*x^2)/(1-4*x+6*x^2-18*x^3+11*x^4-22*x^5 +6*x^6-6*x^7)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 06 2019

(GAP) a:=[1, 4, 16, 58, 208, 750, 2708];; for n in [8..30] do a[n]:=4*a[n-1] -6*a[n-2]+18*a[n-3]-11*a[n-4]+22*a[n-5]-6*a[n-6]+6*a[n-7]; od; a; # G. C. Greubel, Aug 06 2019

CROSSREFS

Cf. A005251, A123892, A123894, A225685.

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane, Nov 20 2006

STATUS

approved

A123894

Expansion of g.f.: (1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)/(1-5*x+10*x^2 -40*x^3+35*x^4-105*x^5 +50*x^6-100*x^7+24*x^8-24*x^9).

+10
4

1, 5, 25, 115, 525, 2405, 11025, 50525, 231525, 1060979, 4862065, 22280875, 102104045, 467901065, 2144199765, 9825991725, 45028504525, 206347239279, 945605099381, 4333321849875, 19857843661645, 91000384646845, 417017585036185, 1911021221349165

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Number of words of length n over {0,1,2,3,4} which have no factor iji with i>j. - N. J. A. Sloane, May 21 2013

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001; Annals. Combin., 7 (2003), 1-14.

Index entries for linear recurrences with constant coefficients, signature (5,-10,40,-35,105,-50,100,-24,24).

FORMULA

G.f. may be written more symmetrically as 1/(1-x*(1 +1/(1+x^2) +1/(1+2*x^2) +1/(1+3*x^2) +1/(1+4*x^2))). - N. J. A. Sloane, May 21 2013

MAPLE

seq(coeff(series((1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)/(1 -5*x +10*x^2 -40*x^3+35*x^4-105*x^5+50*x^6-100*x^7+24*x^8-24*x^9), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 06 2019

MATHEMATICA

CoefficientList[1/(1 - x(1 + 1/(1+x^2) + 1/(1+2x^2) + 1/(1+3x^2) + 1/(1+4x^2))) + O[x]^30, x] (* Jean-François Alcover, Jan 09 2019 *)

PROG

(PARI) my(x='x+O('x^30)); Vec((1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)/(1 -5*x +10*x^2 -40*x^3+35*x^4-105*x^5+50*x^6-100*x^7+24*x^8-24*x^9)) \\ G. C. Greubel, Aug 06 2019

(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)/(1-5*x+10*x^2 -40*x^3+35*x^4-105*x^5 +50*x^6-100*x^7+24*x^8-24*x^9) )); // G. C. Greubel, Aug 06 2019

(Sage) ((1+x^2)*(1+2*x^2)*(1+3*x^2)*(1+4*x^2)/(1-5*x+10*x^2-40*x^3 +35*x^4-105*x^5 +50*x^6-100*x^7+24*x^8-24*x^9)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 06 2019

(GAP) a:=[1, 5, 25, 115, 525, 2405, 11025, 50525, 231525];; for n in [10..30] do a[n]:=5*a[n-1]-10*a[n-2] +40*a[n-3]-35*a[n-4]+105*a[n-5] -50*a[n-6]+100*a[n-7]-24*a[n-8]+24*a[n-9]; od; a; # G. C. Greubel, Aug 06 2019

CROSSREFS

Cf. A005251, A123892, A123893, A225685.

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane, Nov 20 2006

STATUS

approved

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