A265247 -id:A265247 - cp's OEIS frontend

A265248 Sum of the 2nd smallest parts of all the partitions of n (2nd smallest part is defined to be 0 when the partition does not have at least 2 distinct parts).

0, 0, 2, 5, 14, 22, 43, 63, 97, 140, 201, 266, 371, 492, 638, 837, 1079, 1377, 1748, 2207, 2756, 3471, 4287, 5317, 6537, 8081, 9840, 12069, 14643, 17837, 21543, 26113, 31385, 37877, 45318, 54433, 64944, 77682, 92341, 109995, 130373, 154769, 182866, 216350, 254905, 300648, 353259, 415392, 486843, 570867

Offset: 1

Emeric Deutsch, Dec 24 2015

nonn

a(n) = Sum_{k>=0} k*A265247(n,k).

			a(4) = 5 because in [4], [3,1], [2,2], [2,1,1], [1,1,1,1] the 2nd smallest parts are 0,3,0,2,0, respectively.

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000 (terms 1..1000 from Alois P. Heinz)

Cf. A265247.

g := add(x^i*add(j*x^j/mul(1-x^k, k = j .. 100), j = i+1 .. 100)/(1-x^i), i = 1 .. 100): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 1 .. 50);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0],
      `if`(i>n, 0, add((p-> `if`(t=1, p+[0, i*p[1]], p))(
       b(n-i*j, i+1, min(t+1,2))), j=1..n/i)+b(n, i+1, t)))
    end:
a:= n-> b(n,1,0)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 31 2015

Table[Total@ Flatten@ Map[Take[DeleteDuplicates@ #, {-2}] &, Select[IntegerPartitions@ n, Total@ Differences@ # != 0 && Length@ # >= 2 &]], {n, 50}] (* Michael De Vlieger, Dec 24 2015 *)
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0},
     If[i > n, {0, 0}, Sum[If[t == 1, # + {0, i*#[[1]]}, #]&[
     b[n - i*j, i+1, Min[t+1, 2]]], {j, 1, n/i}] + b[n, i+1, t]]];
a[n_] := b[n, 1, 0][[2]];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz *)

G.f.: G(x) = Sum_{i>=1} x^i/(1-x^i)*Sum_{j>=i+1} j*x^j/Product_{k>=j}(1-x^k).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*sqrt(3)*n). - Vaclav Kotesovec, Jun 12 2025

A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Emeric Deutsch, Dec 06 2015

Number of entries in row n = 1 + n^2.
Sum of entries in row n = A000041(n).
Sum(k*T(n,k), k>=0) = A066183(n).

			Row 3 is 0,0,0,1,0,1,0,0,0,1 because in the partitions of 3, namely [1,1,1], [2,1], [3], the sums of the squares of the parts are 3, 5, and 9, respectively.
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,1,0,0,0,1;
0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1.

Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Cf. A000041, A066183, A229325 - A229332, A264402, A265247 - A265253.

g := 1/(product(1-t^(k^2)*x^k, k = 1 .. 100)): gser := simplify(series(g, x = 0, 15)): for n from 0 to 8 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 8 do seq(coeff(P[n], t, j), j = 0 .. n^2) end do; # yields sequence in triangular form

m = 8; CoefficientList[#, t]& /@ CoefficientList[1/Product[(1 - t^(k^2)* x^k), {k, 1, m}] + O[x]^m, x] // Flatten (* Jean-François Alcover, Feb 19 2019 *)

G.f.: G(t,x) = 1/Product_{k>=1} (1 - t^{k^2}*x^k).

cp's OEIS Frontend

A265248 Sum of the 2nd smallest parts of all the partitions of n (2nd smallest part is defined to be 0 when the partition does not have at least 2 distinct parts).

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