A001993 Number of two-rowed partitions of length 3.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Programs

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 *)

Formula

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

Extensions

More terms from James Sellers, Feb 09 2000