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A001993
Number of two-rowed partitions of length 3.
9
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
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)).
a(n) = floor((6*n^5 + 225*n^4 + 3160*n^3 + 20475*n^2 + 58629*n + 103680)/103680 + (-1)^n*(n^2+15*n)/256 + ((2*n^2+1) mod 3)*n/27). - Hoang Xuan Thanh, Feb 14 2026
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
KEYWORD
nonn,easy
EXTENSIONS
More terms from James Sellers, Feb 09 2000
STATUS
approved