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a(n) = sum of the origin-to-boundary graph-distances of all partitions of n.
2

%I #4 Jan 12 2024 16:33:17

%S 0,1,2,4,8,12,21,32,50,73,107,152,219,302,419,567,771,1027,1374,1806,

%T 2375,3083,3999,5136,6597,8398,10676,13477,16981,21260,26584,33057,

%U 41049,50738,62605,76930,94374,115330,140704,171106,207732,251460,303919,366335,440880,529298

%N a(n) = sum of the origin-to-boundary graph-distances of all partitions of n.

%C The origin-to-boundary graph-distance (see A325188) is the side length of the maximum triangular partition contained inside the Ferrer's diagram of the partition. a(n) is the sum of the side lengths over all partitions of n.

%F a(n) = Sum_{k=1..n} k*A325188(n,k).

%o (PARI) a(n)={my(s=0); forpart(p=n, my(w=#p); for(i=1, #p, w=min(w, #p-i+p[i])); s += w); s}

%Y Cf. A325188, A325189, A325200.

%K nonn

%O 0,3

%A _Andrew Howroyd_, Jan 12 2024