A001996 Number of partitions of n into parts 2, 3, 4, 5, 6, 7. (Formerly M0306 N0112)

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 17, 23, 26, 33, 37, 47, 52, 64, 72, 86, 96, 115, 127, 149, 166, 192, 212, 245, 269, 307, 338, 382, 419, 472, 515, 576, 629, 699, 760, 843, 913, 1007, 1091, 1197, 1293, 1416, 1525, 1663, 1790, 1945, 2088, 2265, 2426

Offset

0,5

Comments

Also, Molien series for invariants of finite Coxeter group A_6. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1-x^i). - N. J. A. Sloane, Jan 11 2016

Cayley tabulates the coefficients in the expansion of H = 1 / ((1 - x^2) * (1 - x^4) * ... * (1 - x^14)) with even indices 0, 2, ..., 142.

References

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.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

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

T. D. Noe, Table of n, a(n) for n = 0..1000

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, American Journal of Mathematics, 2 (1879), pp.71-84. See pp.77-78.

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]

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

Formula

G.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

Euler transform of length 7 sequence [ 0, 1, 1, 1, 1, 1, 1]. - Michael Somos, Apr 23 2014

Example

G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
G.f. = 1 + q^2 + q^6 + 2*q^8 + 2*q^10 + 4*q^12 + 4*q^14 + 6*q^16 + ...

Mathematica

nn = 102; t = CoefficientList[Series[1/((1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^10)*(1 - x^12)*(1 - x^14)), {x, 0, nn}], x]; t = Take[t, {1, nn, 2}]

Crossrefs

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

Sequence in context: A241317 A357456 A185224 * A317084 A122134 A035940

Adjacent sequences: A001993 A001994 A001995 * A001997 A001998 A001999

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Feb 09 2000

Status

approved