A002889 -id:A002889 - cp's OEIS frontend

A046741 Triangle read by rows giving number of arrangements of k dumbbells on 2 X n grid (n >= 0, k >= 0).

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

Offset: 0

N. J. A. Sloane

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

			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;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
D. G. Rogers, An application of renewal sequences to the dimer problem, pp. 142-153 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.
Eric Weisstein's World of Mathematics, Ladder Graph
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Donovan Young, The Number of Domino Matchings in the Game of Memory, J. Int. Seq., Vol. 21 (2018), Article 18.8.1.
Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

Diagonals give A002940, A002941, A002889.

Row sums yield A030186. T(n,n) = Fibonacci(n+1) (A000045).

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))
-- Reinhard Zumkeller, Jan 18 2014

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

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 *)

From Emeric Deutsch, Dec 25 2004: (Start)
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).

More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000

Formula fixed by Reinhard Zumkeller, Jan 18 2014

A002940 Arrays of dumbbells.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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 j1). 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
Partial Sums of A023610 - John Molokach, Jul 03 2013

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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Carlos Alirio Rico Acevedo and Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 23-24.
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A002941, A002889, A055608, A062123, A062124, A062125, A062126, A062127, A046741.

Cf. A000045, A001925, A023610, A054454, A006478, A067331, A178523.

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)
-- Reinhard Zumkeller, Jan 18 2014

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/((1-x)*(1-x-x^2)^2) )); // G. C. Greubel, Jan 31 2019

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 *)

my(x='x+O('x^35)); Vec((1+x)/((1-x)*(1-x-x^2)^2)) \\ G. C. Greubel, Jan 31 2019

((1+x)/((1-x)*(1-x-x^2)^2)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

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

More terms from Henry Bottomley, Jun 02 2000

A002941 Arrays of dumbbells.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (5,-7,-2,10,-2,-5,1,1).

Cf. A046741, A002940, A002889, A055608, A062123-A062127.

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)
-- Reinhard Zumkeller, Jan 18 2014

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^2/((1-x-x^2)^3*(1-x)^2) )); // G. C. Greubel, Jan 31 2019

CoefficientList[(1+x)^2/((1-x-x^2)^3*(1-x)^2) + O[x]^30, x] (* Jean-François Alcover, Jul 31 2018 *)
LinearRecurrence[{5,-7,-2,10,-2,-5,1,1},{1,7,29,94,263,667,1577,3538},30] (* Harvey P. Dale, Aug 29 2021 *)

x='x+O('x^30); Vec((1+x)^2/((1-x-x^2)^3*(1-x)^2)) \\ Altug Alkan, Jul 31 2018

((1+x)^2/((1-x-x^2)^3*(1-x)^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

G.f.: (1+x)^2/((1-x-x^2)^3*(1-x)^2).

a(n) = 2*a(n-1) - a(n-3) + A002940(n) + A002940(n-1).

More terms from Henry Bottomley, Jun 02 2000

A062123 a(n) = (9n^2 + 9n + 4)/2.

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Jun 04 2001

Third column of A046741.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062124-A062127.

Cf. A235332.

List([0..50], n -> 2 +9*n*(1+n)/2); # G. C. Greubel, Jan 31 2019

[2 +9*n*(1+n)/2: n in [0..50]]; // G. C. Greubel, Jan 31 2019

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 *)

for (n=0, 1000, write("b062123.txt", n, " ", 2 + (n + n^2)*9/2) ) \\ Harry J. Smith, Aug 02 2009

[2 +9*n*(1+n)/2 for n in range(50)] # G. C. Greubel, Jan 31 2019

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).
a(n) = 9*n + a(n-1), with n>0, a(0)=2. - Vincenzo Librandi, Aug 07 2010
E.g.f.: (4 +18*x +9*x^2)*exp(x)/2. - G. C. Greubel, Jan 31 2019

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A055608 Arrays of dumbbells.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jun 02 2000

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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-31,44,4,-84,66,46,-74,-4,36,-4,-9,1,1).

Cf. A002940, A002941, A002889, A046741.

Cf. A062123-A062127.

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)
-- Reinhard Zumkeller, Jan 18 2014

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^4/((1-x)^4*(1-x-x^2)^5) )); // G. C. Greubel, Jan 31 2019

CoefficientList[Series[(1+x)^4/((1-x)^4*(1-x-x^2)^5), {x,0,30}], x] (* G. C. Greubel, Jan 31 2019 *)

my(x='x+O('x^30)); Vec((1+x)^4/((1-x)^4*(1-x-x^2)^5)) \\ G. C. Greubel, Jan 31 2019

((1+x)^4/((1-x)^4*(1-x-x^2)^5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

G.f.: (1+x)^4/((1-x)^4*(1-x-x^2)^5).

a(n) = 2*a(n-1) - a(n-3) + A002889(n) + A002889(n-1).

A062127 Seventh column of A046741.

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

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

[(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

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 *)

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

[(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

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).
From G. C. Greubel, Jan 31 2019: (Start)
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)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A062124 Fourth column of A046741.

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (6+19*n+18*n^2+9*n^3)/2); # G. C. Greubel, Jan 31 2019

[(6+19*n+18*n^2+9*n^3)/2: n in [0..40]]; // G. C. Greubel, Jan 31 2019

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 *)

vector(40, n, n--; (6+19*n+18*n^2+9*n^3)/2) \\ G. C. Greubel, Jan 31 2019

[(6+19*n+18*n^2+9*n^3)/2 for n in range(40)] # G. C. Greubel, Jan 31 2019

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).
From G. C. Greubel, Jan 31 2019: (Start)
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)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A062125 Fifth column of A046741.

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (40+126*n+165*n^2+90*n^3+27*n^4)/8); # G. C. Greubel, Jan 31 2019

[(40+126*n+165*n^2+90*n^3+27*n^4)/8: n in [0..40]]; // G. C. Greubel, Jan 31 2019

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 *)

vector(40, n, n--; (40+126*n+165*n^2+90*n^3+27*n^4)/8) \\ G. C. Greubel, Jan 31 2019

[(40+126*n+165*n^2+90*n^3+27*n^4)/8 for n in range(40)] # G. C. Greubel, Jan 31 2019

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
From G. C. Greubel, Jan 31 2019: (Start)
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)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A062126 Sixth column of A046741.

Original entry on oeis.org

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

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

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

[(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

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 *)

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

[(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

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).
From G. C. Greubel, Jan 31 2019: (Start)
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)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A123518 Number of dumbbells in all possible arrangements of dumbbells on a 2 X n rectangular array of compartments.

Original entry on oeis.org

1, 8, 38, 166, 671, 2602, 9792, 36068, 130697, 467556, 1655406, 5811290, 20255279, 70172502, 241839184, 829685064, 2835099649, 9653650752, 32768012102, 110913651342, 374469646511, 1261386990850, 4240037471152, 14225209349036

Offset: 1

Emeric Deutsch, Oct 16 2006

nonn

			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.

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
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.
Index entries for linear recurrences with constant coefficients, signature (6,-7,-8,5,2,-1).

Cf. A030186, A046741.

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

R:=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

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);

LinearRecurrence[{6,-7,-8,5,2,-1}, {1,8,38,166,671,2602}, 30] (* G. C. Greubel, Oct 28 2019 *)

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

def A123518_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x-3*x^2+2*x^3)/(1-3*x-x^2+x^3)^2 ).list()
a=A123518_list(30); a[1:] # G. C. Greubel, Oct 28 2019

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.

cp's OEIS Frontend

A046741 Triangle read by rows giving number of arrangements of k dumbbells on 2 X n grid (n >= 0, k >= 0).

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A002940 Arrays of dumbbells.

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A002941 Arrays of dumbbells.

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A062123 a(n) = (9n^2 + 9n + 4)/2.

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A055608 Arrays of dumbbells.

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A062127 Seventh column of A046741.

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