A002043 MacMahon's solid partitions of n in which 2 is the smallest summand. (Formerly M2204 N1710)

3, 0, 6, 18, 40, 81, 201, 414, 916, 1899, 3973, 8059, 16402, 32561, 64520, 125986, 244448, 469195, 895077, 1692143, 3179406, 5929721, 10993373, 20250589, 37096872, 67568512, 122437970, 220721343, 395998810, 707122884, 1257092767, 2225113351, 3922306946

Offset

2,1

References

R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139.

V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314.

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

David Radcliffe, Table of n, a(n) for n = 2..1000

R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139. [Annotated scanned copy]

V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314. [Annotated scanned copy]

Formula

a(n) = 3*(p(n-2)-p(n-3)-p(n-4)+p(n-5))+p(n-6)-p(n-7), where p(n)=A000294(n) for n>=0 and p(n)=0 for n<0. - David Radcliffe, Jan 03 2026

Prog

(PARI) a(n, m=2)=Vec(prod(r=m, n, 1/(1-'x^r+O('x^(n+1)))^binomial(r+1, 2)-(r==m))); /* Martin Fuller, Jan 03 2026 */
(Python)
from functools import cache
from sympy import divisor_sigma as s
@cache
def p(n): return 0 if n<0 else 1 if n==0 else sum((s(m, 2)+s(m, 3))*p(n-m) for m in range(1, n+1))/(2*n)
def a(n): return 3*(p(n-2)-p(n-3)-p(n-4)+p(n-5))+p(n-6)-p(n-7) # David Radcliffe, Jan 03 2026

Crossrefs

Cf. A000294.

Sequence in context: A059684 A270509 A083350 * A171002 A375987 A137436

Adjacent sequences: A002040 A002041 A002042 * A002044 A002045 A002046

Keyword

nonn

Author

N. J. A. Sloane

Extensions

a(27)-a(34) added and name made more specific by David Radcliffe, Jan 03 2026

Status

approved