A168606 The number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly four nonempty parts.

1, 4, 20, 102, 496, 2294, 10200, 44062, 186416, 776934, 3203080, 13101422, 53279136, 215749174, 870919160, 3507493182, 14101520656, 56620923014, 227128606440, 910449955342, 3647607982976, 14607859562454, 58483727432920

Offset: 4

Author

Martin Griffiths, Dec 01 2009

Keywords

• nonn
• easy

Comments

The number of ways of partitioning the multiset {1, 1, 1, 2, 3, ..., n-1} into exactly two and three nonempty parts are given in A168604 and A168605 respectively.

Links

• G. C. Greubel, Table of n, a(n) for n = 4..1000
• M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
• Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Crossrefs

Cf. A168604, A168605.

Programs

Magma

[(10*4^(n-4) -5*3^(n-3) +9*2^(n-4) -1)/3: n in [4..30]]; // G. C. Greubel, Feb 07 2021

Mathematica

a[n_]:= (10*4^(n-4) - 5*3^(n-3) + 9*2^(n-4) - 1)/3; Table[a[n], {n, 4, 30}]

Sage

[(10*4^(n-4) -5*3^(n-3) +9*2^(n-4) -1)/3 for n in (4..30)] # G. C. Greubel, Feb 07 2021

Formula

a(n) = (10*4^(n-4) - 5*3^(n-3) + 9*2^(n-4) - 1)/3.

The shifted e.g.f. is (10*exp(4*x) - 15*exp(3*x) + 9*exp(2*x) - exp(x))/3.

G.f.: x^4*(1 -6*x +15*x^2 -8*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).

Extensions

Last element of the multiset in the definition corrected by Martin Griffiths, Dec 02 2009