cp's OEIS Frontend

A238641 Number of partitions p of 2n-1 such that n - (number of parts of p) is a part of p.

%I A238641 #14 Jun 28 2021 23:09:36
%S A238641 0,0,1,4,9,21,37,69,113,187,286,449,657,976,1397,2003,2788,3902,5323,
%T A238641 7284,9789,13144,17405,23052,30142,39379,50967,65842,84368,107954,
%U A238641 137126,173893
%N A238641 Number of partitions p of 2n-1 such that n - (number of parts of p) is a part of p.
%H A238641 Giovanni Resta, <a href="/A238641/b238641.txt">Table of n, a(n) for n = 1..1000</a>
%H A238641 Giulio Ruzza and Di Yang, <a href="https://arxiv.org/abs/2104.01480">On the spectral problem of the quantum KdV hierarchy</a>, arXiv:2104.01480 [math-ph], 2021.
%e A238641 a(4) counts these partitions of 7:  52, 511, 421, 331.
%t A238641 z = 30; g[n_] := IntegerPartitions[n]; m[p_, t_] := MemberQ[p, t];
%t A238641 Table[Count[g[2 n], p_ /; m[p, n - Length[p]]], {n, z}] (*A238607*)
%t A238641 Table[Count[g[2 n - 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238641*)
%t A238641 Table[Count[g[2 n + 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238742*)
%t A238641 p[n_, k_] := p[n, k] = If[k == 1 || n == k, 1, If[k > n, 0, p[n - 1, k - 1] + p[n - k, k]]]; q[n_, k_, e_] := q[n, k, e] = If[n - e < k - 1 , 0, If[k == 1, If[n == e, 1, 0], p[n - e, k - 1]]]; a[n_] := Sum[q[2*n - 1, u, n - u], {u, n - 1}]; Array[a,100] (* _Giovanni Resta_, Mar 09 2014 *)
%Y A238641 Cf. A238640, A238742.
%K A238641 nonn,easy
%O A238641 1,4
%A A238641 _Clark Kimberling_, Mar 04 2014