A001993 Number of two-rowed partitions of length 3. (Formerly M2452 N0973)

1, 1, 3, 5, 9, 13, 22, 30, 45, 61, 85, 111, 150, 190, 247, 309, 390, 478, 593, 715, 870, 1038, 1243, 1465, 1735, 2023, 2368, 2740, 3175, 3643, 4189, 4771, 5443, 6163, 6982, 7858, 8852, 9908, 11098, 12366, 13780, 15284, 16958, 18730, 20692, 22772, 25058, 27478

Offset

0,3

References

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.

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).

Links

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

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]

L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.

Index entries for linear recurrences with constant coefficients, signature (1, 2, 0, -2, -4, 1, 3, 3, 1, -4, -2, 0, 2, 1, -1).

Formula

G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)).

Maple

a:= n-> (Matrix(15, (i, j)-> if (i=j-1) then 1 elif j=1 then [1, 2, 0, -2, -4, 1, 3, 3, 1, -4, -2, 0, 2, 1, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 31 2008

Mathematica

a[n_] := (Table[Which[i == j-1, 1, j == 1, {1, 2, 0, -2, -4, 1, 3, 3, 1, -4, -2, 0, 2, 1, -1}[[i]], True, 0], {i, 1, 15}, {j, 1, 15}] // MatrixPower[#, n]&)[[1, 1]]; Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

Crossrefs

Sequence in context: A248604 A146905 A052282 * A284829 A153263 A295140

Adjacent sequences: A001990 A001991 A001992 * A001994 A001995 A001996

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Feb 09 2000

Status

approved