A099948 Number of partitions of n such that the number of blocks is congruent to 3 mod 4.

1, 6, 25, 90, 302, 994, 3487, 15210, 92489, 713988, 5979480, 50184316, 412595913, 3317961318, 26241631409, 205918294518, 1622545217510, 13045429410974, 109152638729439, 969395726250226, 9255388478615017, 94973500733767432, 1034488089509527120

Offset

3,2

Links

Alois P. Heinz, Table of n, a(n) for n = 3..500

M. Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.

Formula

G.f.: sum(x^k/[(1-x)(1-2x)...(1-kx)], k=3 (mod 4)). - Emeric Deutsch, Dec 15 2004

Example

a(11)=92489 because stirling2(11,3)+stirling2(11,7)+stirling2(11,11)=92489.

Maple

seq(sum(stirling2(n, 3+4*k), k=0..(n-3)/4), n=3..26); # Emeric Deutsch, Dec 15 2004
# second Maple program:
with(combinat):
b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=3, 1, 0),
     `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1, irem(m+j, 4)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=3..30);  # Alois P. Heinz, Sep 17 2015

Mathematica

Table[Sum[StirlingS2[n, 3+4*k], {k, 0, (n-3)/4}], {n, 3, 26}] (* Jean-François Alcover, Feb 18 2016, after Emeric Deutsch *)

Crossrefs

Cf. A143817, A358499.

Sequence in context: A001871 A000392 A365531 * A333017 A277973 A143815

Adjacent sequences: A099945 A099946 A099947 * A099949 A099950 A099951

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 12 2004

Extensions

More terms from Emeric Deutsch, Dec 15 2004

Status

approved