| Displaying 1-10 of 10 results found.
page
1
Triangle read by rows giving number of arrangements of k dumbbells on 2 X n grid (n >= 0, k >= 0).
+10
25
1, 1, 1, 1, 4, 2, 1, 7, 11, 3, 1, 10, 29, 26, 5, 1, 13, 56, 94, 56, 8, 1, 16, 92, 234, 263, 114, 13, 1, 19, 137, 473, 815, 667, 223, 21, 1, 22, 191, 838, 1982, 2504, 1577, 424, 34, 1, 25, 254, 1356, 4115, 7191, 7018, 3538, 789, 55, 1, 28, 326, 2054, 7646, 17266, 23431
COMMENTS
Equivalently, T(n,k) is the number of k-matchings in the ladder graph L_n = P_2 X P_n. - Emeric Deutsch, Dec 25 2004 In other words, triangle of number of monomer-dimer tilings on (2,n)-block with k dimers. If z marks the size of the block and t marks the dimers, then it is easy to see that the g.f. for indecomposable tilings, i.e., those that cannot be split vertically into smaller tilings, is g = (1+t)*z + t^2*z^2 + 2*t*z^2 + 2*t^2*z^3 + 2*t^3*z^4 + ... = (1+t)*z + t^2*z^2 + 2*t*z^2/(1-t*z); then the g.f. is 1/(1-g) = (1-t*z)/(1 - z - 2*t*z - t*z^2 + t^3*z^3) (see eq. (4) of the Grimson reference). From this the recurrence of the McQuistan & Lichtman reference follows at once. - Emeric Deutsch, Oct 16 2006
FORMULA
The row generating polynomials P[n] satisfy P[n] = (1 + 2*t)*P[n-1] + t*P[n-2] - t^3*P[n-3] with P[0] = 1, P[1] = 1+t, P[2] = 1 + 4*t + 2*t^2.
G.f.: (1-t*z)/(1 - z - 2*t*z - t*z^2 + t^3*z^3). (End)
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-3).
EXAMPLE
T(3, 2)=11 because in the 2 X 3 grid with vertex set {O(0, 0), A(1, 0), B(2, 0), C(2, 1), D(1, 1), E(0, 1)} and edge set {OA, AB, ED, DC, UE, AD, BC} we have the following eleven 2-matchings: {OA, BC}, {OA, DC}, {OA, ED}, {AB, DC}, {AB, ED}, {AB, OE}, {BC, AD}, {BC, ED}, {BC, OA}, {BC, OE} and {DC, OE}. - Emeric Deutsch, Dec 25 2004 Triangle starts:
1;
1, 1;
1, 4, 2;
1, 7, 11, 3;
1, 10, 29, 26, 5;
MAPLE
F[0]:=1:F[1]:=1+t:F[2]:=1+4*t+2*t^2:for n from 3 to 10 do F[n]:=sort(expand((1+2*t)*F[n-1]+t*F[n-2]-t^3*F[n-3])) od: for n from 0 to 10 do seq(coeff(t*F[n], t^k), k=1..n+1) od; # yields sequence in triangular form - Emeric Deutsch
MATHEMATICA
p[n_] := p[n] = (1 + 2t) p[n-1] + t*p[n-2] - t^3*p[n-3]; p[0] = 1; p[1] = 1+t; p[2] = 1 + 4t + 2t^2; Flatten[Table[CoefficientList[Series[p[n], {t, 0, n}], t], {n, 0, 10}]][[;; 62]] (* Jean-François Alcover, Jul 13 2011, after Emeric Deutsch *) CoefficientList[LinearRecurrence[{1 + 2 x, x, -x^3}, {1 + x, 1 + 4 x + 2 x^2, 1 + 7 x + 11 x^2 + 3 x^3}, {0, 10}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *) CoefficientList[CoefficientList[Series[-(1 + x z) (-1 - x + x^2 z)/(1 - z - 2 x z - x z^2 + x^3 z^3), {z, 0, 10}], z], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
PROG
(Haskell)
a046741 n k = a046741_tabl !! n !! k
a046741_row n = a046741_tabl !! n
a046741_tabl = [[1], [1, 1], [1, 4, 2]] ++ f [1] [1, 1] [1, 4, 2] where
f us vs ws = ys : f vs ws ys where
ys = zipWith (+) (zipWith (+) (ws ++ [0]) ([0] ++ map (* 2) ws))
(zipWith (-) ([0] ++ vs ++ [0]) ([0, 0, 0] ++ us))
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000
Arrays of dumbbells.
(Formerly M3415 N1381)
+10
17
1, 4, 11, 26, 56, 114, 223, 424, 789, 1444, 2608, 4660, 8253, 14508, 25343, 44030, 76136, 131110, 224955, 384720, 656041, 1115784, 1893216, 3205416, 5416441, 9136084, 15384563, 25866914, 43429784, 72821274, 121953943, 204002680, 340886973, 569047468, 949022608
COMMENTS
Whitney transform of n. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)). - Paul Barry, Feb 16 2005 a(n-1) is the permanent of the n X n 0-1 matrix with 1 in (i,j) position iff (i=1 and j<n) or 0<=i-j<=2 or (j=n and i>1). For example, with n=5, a(4) = per([[1, 1, 1, 1, 0], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1]]) = 26. - David Callan, Jun 07 2006 a(n) is the internal path length of the Fibonacci tree of order n+2. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. The internal path length of a tree is the sum of the levels of all of its internal (i.e. non-leaf) nodes. - Emeric Deutsch, Jun 15 2010
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
FORMULA
a(n) = 2*a(n-1) - a(n-3) + A000045(n+1). G.f.: x*(1+x)/((1-x)*(1-x-x^2)^2).
a(n) = Sum_{k=0..n} ( Sum_{i=0..n} k*C(k, i-k) ). - Paul Barry, Feb 16 2005 E.g.f.: 2*exp(x) + exp(x/2)*((55*x - 50)*cosh(sqrt(5)*x/2) + sqrt(5)*(25*x - 22)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 03 2023
MATHEMATICA
a[n_]:= a[n]= If[n<3, n^2, 2a[n-1] -a[n-3] +Fibonacci[n+1]]; Array[a, 32] (* Jean-François Alcover, Jul 31 2018 *)
PROG
(Haskell)
a002940 n = a002940_list !! (n-1)
a002940_list = 1 : 4 : 11 : zipWith (+)
(zipWith (-) (map (* 2) $ drop 2 a002940_list) a002940_list)
(drop 5 a000045_list)
(PARI) my(x='x+O('x^35)); Vec((1+x)/((1-x)*(1-x-x^2)^2)) \\ G. C. Greubel, Jan 31 2019 (Magma) m:=35; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/((1-x)*(1-x-x^2)^2) )); // G. C. Greubel, Jan 31 2019 (Sage) ((1+x)/((1-x)*(1-x-x^2)^2)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019
Arrays of dumbbells.
(Formerly M4396 N1852)
+10
12
1, 7, 29, 94, 263, 667, 1577, 3538, 7622, 15900, 32314, 64274, 125561, 241569, 458715, 861242, 1601081, 2950693, 5396209, 9801012, 17692092, 31759800, 56727588, 100861716, 178585489, 314995915, 553650761, 969967510, 1694235803
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
G.f.: (1+x)^2/((1-x-x^2)^3*(1-x)^2).
MATHEMATICA
LinearRecurrence[{5, -7, -2, 10, -2, -5, 1, 1}, {1, 7, 29, 94, 263, 667, 1577, 3538}, 30] (* Harvey P. Dale, Aug 29 2021 *)
PROG
(Haskell)
a002941 n = a002941_list !! (n-1)
a002941_list = 1 : 7 : 29 : zipWith (+)
(zipWith (-) (map (* 2) $ drop 2 a002941_list) a002941_list)
(drop 2 $ zipWith (+) (tail a002940_list) a002940_list)
(PARI) x='x+O('x^30); Vec((1+x)^2/((1-x-x^2)^3*(1-x)^2)) \\ Altug Alkan, Jul 31 2018 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^2/((1-x-x^2)^3*(1-x)^2) )); // G. C. Greubel, Jan 31 2019 (Sage) ((1+x)^2/((1-x-x^2)^3*(1-x)^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019
a(n) = (9n^2 + 9n + 4)/2.
+10
10
2, 11, 29, 56, 92, 137, 191, 254, 326, 407, 497, 596, 704, 821, 947, 1082, 1226, 1379, 1541, 1712, 1892, 2081, 2279, 2486, 2702, 2927, 3161, 3404, 3656, 3917, 4187, 4466, 4754, 5051, 5357, 5672, 5996, 6329, 6671, 7022, 7382, 7751, 8129, 8516, 8912, 9317
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
FORMULA
G.f.: (1+2*x)*(2+x)/(1-x)^3. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y)-(1+y)*x).
MATHEMATICA
Table[2 +9*n*(1+n)/2, {n, 0, 50}] (* G. C. Greubel, Jan 31 2019 *) LinearRecurrence[{3, -3, 1}, {2, 11, 29}, 50] (* Harvey P. Dale, Jan 12 2020 *)
PROG
(PARI) for (n=0, 1000, write("b062123.txt", n, " ", 2 + (n + n^2)*9/2) ) \\ Harry J. Smith, Aug 02 2009 (Magma) [2 +9*n*(1+n)/2: n in [0..50]]; // G. C. Greubel, Jan 31 2019 (Sage) [2 +9*n*(1+n)/2 for n in range(50)] # G. C. Greubel, Jan 31 2019 (GAP) List([0..50], n -> 2 +9*n*(1+n)/2); # G. C. Greubel, Jan 31 2019
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
1, 13, 92, 473, 1982, 7191, 23431, 70234, 196941, 522939, 1327002, 3240917, 7660538, 17602967, 39466363, 86593478, 186399956, 394478234, 822229746, 1690521204, 3433033150, 6893852746, 13702694284, 26982983126, 52680389239
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
LINKS
Index entries for linear recurrences with constant coefficients, signature (9,-31,44,4,-84,66,46,-74,-4,36,-4,-9,1,1).
FORMULA
G.f.: (1+x)^4/((1-x)^4*(1-x-x^2)^5).
MATHEMATICA
CoefficientList[Series[(1+x)^4/((1-x)^4*(1-x-x^2)^5), {x, 0, 30}], x] (* G. C. Greubel, Jan 31 2019 *)
PROG
(Haskell)
a055608 n = a055608_list !! (n-1)
a055608_list = 1 : 13 : 92 : zipWith (+)
(zipWith (-) (map (* 2) $ drop 2 a055608_list) a055608_list)
(drop 2 $ zipWith (+) (tail a002889_list) a002889_list)
(PARI) my(x='x+O('x^30)); Vec((1+x)^4/((1-x)^4*(1-x-x^2)^5)) \\ G. C. Greubel, Jan 31 2019 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^4/((1-x)^4*(1-x-x^2)^5) )); // G. C. Greubel, Jan 31 2019 (Sage) ((1+x)^4/((1-x)^4*(1-x-x^2)^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019
13, 223, 1577, 7018, 23431, 64316, 153190, 327718, 644573, 1185025, 2061259, 3423422, 5467399, 8443318, 12664784, 18518842, 26476669, 37104995, 51078253, 69191458, 92373815, 121703056, 158420506, 203946878, 259898797, 328106053
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).
FORMULA
G.f.: (2*x^6 + 14*x^5 + 72*x^4 + 207*x^3 + 289*x^2 + 132*x + 13)/(1-x)^7. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y)-(1+y)*x). a(n) = (81*n^6 + 567*n^5 + 2205*n^4 + 4545*n^3 + 5674*n^2 + 3728*n + 1040)/80.
E.g.f.: (1040 + 16800*x + 45760*x^2 + 39240*x^3 + 13140*x^4 + 1782*x^5 + 81*x^6)*exp(x)/80. (End)
MATHEMATICA
Table[(81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80, {n, 0, 40}] (* G. C. Greubel, Jan 31 2019 *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {13, 223, 1577, 7018, 23431, 64316, 153190}, 30] (* Harvey P. Dale, Jun 07 2022 *)
PROG
(PARI) vector(40, n, n--; (81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80) \\ G. C. Greubel, Jan 31 2019 (Magma) [(81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80: n in [0..40]]; // G. C. Greubel, Jan 31 2019 (Sage) [(81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80 for n in range(40)] # G. C. Greubel, Jan 31 2019 (GAP) List([0..40], n -> (81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80); # G. C. Greubel, Jan 31 2019
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
3, 26, 94, 234, 473, 838, 1356, 2054, 2959, 4098, 5498, 7186, 9189, 11534, 14248, 17358, 20891, 24874, 29334, 34298, 39793, 45846, 52484, 59734, 67623, 76178, 85426, 95394, 106109, 117598, 129888, 143006, 156979, 171834, 187598, 204298
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).
FORMULA
G.f.: (3 + 14*x + 8*x^2 + 2*x^3)/(1-x)^4. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y) - (1+y)*x). a(n) = (6 + 19*n + 18*n^2 + 9*n^3)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: (6 + 46*x + 45*x^2 + 9*x^3)*exp(x)/2. (End)
MATHEMATICA
Table[(6+19*n+18*n^2+9*n^3)/2, {n, 0, 40}] (* G. C. Greubel, Jan 31 2019 *) LinearRecurrence[{4, -6, 4, -1}, {3, 26, 94, 234}, 40] (* Harvey P. Dale, Feb 20 2022 *)
PROG
(PARI) vector(40, n, n--; (6+19*n+18*n^2+9*n^3)/2) \\ G. C. Greubel, Jan 31 2019 (Magma) [(6+19*n+18*n^2+9*n^3)/2: n in [0..40]]; // G. C. Greubel, Jan 31 2019 (Sage) [(6+19*n+18*n^2+9*n^3)/2 for n in range(40)] # G. C. Greubel, Jan 31 2019 (GAP) List([0..40], n -> (6+19*n+18*n^2+9*n^3)/2); # G. C. Greubel, Jan 31 2019
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
5, 56, 263, 815, 1982, 4115, 7646, 13088, 21035, 32162, 47225, 67061, 92588, 124805, 164792, 213710, 272801, 343388, 426875, 524747, 638570, 769991, 920738, 1092620, 1287527, 1507430, 1754381, 2030513, 2338040, 2679257, 3056540
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).
FORMULA
G.f.: (5 + 33*x^2 + 10*x^3 + 31*x + 2*x^4)/(1-x)^5. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y)-(1+y)*x). a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), where a(0)=5, a(1)=56, a(2)=263, a(3)=815, a(4)=1982. - Harvey P. Dale, Dec 21 2011 a(n) = (40 + 126*n + 165*n^2 + 90*n^3 + 27*n^4)/8.
E.g.f.: (40 + 408*x + 624*x^2 + 252*x^3 + 27*x^4)*exp(x)/8. (End)
MATHEMATICA
LinearRecurrence[{5, -10, 10, -5, 1}, {5, 56, 263, 815, 1982}, 31] (* or *) CoefficientList[Series[(5+33x^2+10x^3+31x+2x^4)/(1-x)^5, {x, 0, 30}], x] (* Harvey P. Dale, Dec 21 2011 *) Table[(40+126*n+165*n^2+90*n^3+27*n^4)/8, {n, 0, 40}] (* G. C. Greubel, Jan 31 2019 *)
PROG
(PARI) vector(40, n, n--; (40+126*n+165*n^2+90*n^3+27*n^4)/8) \\ G. C. Greubel, Jan 31 2019 (Magma) [(40+126*n+165*n^2+90*n^3+27*n^4)/8: n in [0..40]]; // G. C. Greubel, Jan 31 2019 (Sage) [(40+126*n+165*n^2+90*n^3+27*n^4)/8 for n in range(40)] # G. C. Greubel, Jan 31 2019 (GAP) List([0..40], n -> (40+126*n+165*n^2+90*n^3+27*n^4)/8); # G. C. Greubel, Jan 31 2019
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
8, 114, 667, 2504, 7191, 17266, 36482, 70050, 124882, 209834, 335949, 516700, 768233, 1109610, 1563052, 2154182, 2912268, 3870466, 5066063, 6540720, 8340715, 10517186, 13126374, 16229866, 19894838, 24194298, 29207329
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).
FORMULA
G.f.: (x+2)*(2*x^4 + 8*x^3 + 36*x^2 + 31*x + 4)/(1-x)^6. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1 - y - y^2)*(1-y) - (1+y)*x). a(n) = (320 + 1114*n + 1515*n^2 + 1125*n^3 + 405*n^4 + 81*n^5)/40.
E.g.f.: (320 + 4240*x + 8940*x^2 + 5580*x^3 + 1215*x^4 + 81*x^5)*exp(x)/40. (End)
MATHEMATICA
Table[(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40, {n, 0, 40}] (* G. C. Greubel, Jan 31 2019 *)
PROG
(PARI) vector(40, n, n--; (320+1114*n+1515*n^2+1125*n^3+405*n^4 + 81*n^5)/40) \\ G. C. Greubel, Jan 31 2019 (Magma) [(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40: n in [0..40]]; // G. C. Greubel, Jan 31 2019 (Sage) [(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40 for n in range(40)] # G. C. Greubel, Jan 31 2019 (GAP) List([0..40], n -> (320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5 )/40); # G. C. Greubel, Jan 31 2019
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
Number of dumbbells in all possible arrangements of dumbbells on a 2 X n rectangular array of compartments.
+10
1
1, 8, 38, 166, 671, 2602, 9792, 36068, 130697, 467556, 1655406, 5811290, 20255279, 70172502, 241839184, 829685064, 2835099649, 9653650752, 32768012102, 110913651342, 374469646511, 1261386990850, 4240037471152, 14225209349036
FORMULA
a(n) = Sum_{k=0..n} k* A046741(n,k). G.f.: x*(1 + 2*x - 3*x^2 + 2*x^3)/(1 - 3*x - x^2 + x^3)^2.
EXAMPLE
a(2)=8 because in a 2 X 2 array of compartments, numbered clockwise starting from the NW one, we have 7 (= A030186(2)) possible arrangements of dumbbells: [ ], [14], [23], [12], [34], [14,23] and [12,34] (ij indicates a dumbbell placed in the compartments i and j); these contain altogether 8 dumbbells.
MAPLE
G:=z*(1+2*z-3*z^2+2*z^3)/(1-3*z-z^2+z^3)^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..27);
MATHEMATICA
LinearRecurrence[{6, -7, -8, 5, 2, -1}, {1, 8, 38, 166, 671, 2602}, 30] (* G. C. Greubel, Oct 28 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2) \\ G. C. Greubel, Oct 28 2019 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2 )); // G. C. Greubel, Oct 28 2019 (Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2 ).list()
(GAP) a:=[1, 8, 38, 166, 671, 2602];; for n in [7..30] do a[n]:=6*a[n-1] -7*a[n-2]-8*a[n-3]+5*a[n-4]+2*a[n-5]-a[n-6]; od; a; # G. C. Greubel, Oct 28 2019
Search completed in 0.016 seconds
| 