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Convolution of partition numbers and Bell numbers.
1

%I #18 Jan 09 2023 02:23:11

%S 1,2,5,12,32,95,328,1294,5748,28152,149768,856130,5218107,33712600,

%T 229800588,1646316230,12355374717,96861192976,791258805462,

%U 6720627186126,59234364203973,540812222400025,5106663817693176,49798678281859244,500857393911224861

%N Convolution of partition numbers and Bell numbers.

%H Alois P. Heinz, <a href="/A014326/b014326.txt">Table of n, a(n) for n = 0..500</a>

%p with(combinat):

%p a:= n-> add(numbpart(k)*bell(n-k), k=0..n):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 15 2015

%t a[n_]:= Sum[PartitionsP[k]*BellB[n-k], {k,0,n}];

%t Table[a[n], {n,0,30}] (* _Jean-François Alcover_, Dec 06 2016 *)

%o (Magma)

%o A014326:= func< n | (&+[NumberOfPartitions(j)*Bell(n-j): j in [0..n]]) >;

%o [A014326(n): n in [0..40]]; // _G. C. Greubel_, Jan 08 2023

%o (SageMath)

%o def A014326(n): return sum(number_of_partitions(j)*bell_number(n-j) for j in range(n+1))

%o [A014326(n) for n in range(41)] # _G. C. Greubel_, Jan 08 2023

%Y Cf. A000041, A000110.

%Y Partial sums of A292503.

%K nonn

%O 0,2

%A _N. J. A. Sloane_