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

%I A099948 #16 Nov 24 2022 04:29:54
%S A099948 1,6,25,90,302,994,3487,15210,92489,713988,5979480,50184316,412595913,
%T A099948 3317961318,26241631409,205918294518,1622545217510,13045429410974,
%U A099948 109152638729439,969395726250226,9255388478615017,94973500733767432,1034488089509527120
%N A099948 Number of partitions of n such that the number of blocks is congruent to 3 mod 4.
%H A099948 Alois P. Heinz, <a href="/A099948/b099948.txt">Table of n, a(n) for n = 3..500</a>
%H A099948 M. Klazar, <a href="http://dx.doi.org/10.1016/S0097-3165(03)00014-1">Bell numbers, their relatives and algebraic differential equations</a>, J. Combin. Theory, A 102 (2003), 63-87.
%F A099948 G.f.: sum(x^k/[(1-x)(1-2x)...(1-kx)], k=3 (mod 4)). - _Emeric Deutsch_, Dec 15 2004
%e A099948 a(11)=92489 because stirling2(11,3)+stirling2(11,7)+stirling2(11,11)=92489.
%p A099948 seq(sum(stirling2(n,3+4*k),k=0..(n-3)/4),n=3..26); # _Emeric Deutsch_, Dec 15 2004
%p A099948 # second Maple program:
%p A099948 with(combinat):
%p A099948 b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=3, 1, 0),
%p A099948      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p A099948       b(n-i*j, i-1, irem(m+j, 4)), j=0..n/i)))
%p A099948     end:
%p A099948 a:= n-> b(n$2, 0):
%p A099948 seq(a(n), n=3..30);  # _Alois P. Heinz_, Sep 17 2015
%t A099948 Table[Sum[StirlingS2[n, 3+4*k], {k, 0, (n-3)/4}], {n, 3, 26}] (* _Jean-François Alcover_, Feb 18 2016, after _Emeric Deutsch_ *)
%Y A099948 Cf. A143817, A358499.
%K A099948 nonn,easy
%O A099948 3,2
%A A099948 _N. J. A. Sloane_, Nov 12 2004
%E A099948 More terms from _Emeric Deutsch_, Dec 15 2004