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A102186 The PDO(n) function (Partitions with Designated summands in which all parts are Odd): the sum of products of multiplicities of parts in all partitions of n into odd parts.

%I A102186 #48 Jul 19 2025 13:07:12
%S A102186 1,1,2,4,5,8,12,16,22,32,42,56,76,98,128,168,213,272,348,436,548,688,
%T A102186 852,1056,1308,1603,1964,2404,2920,3544,4296,5176,6230,7488,8958,
%U A102186 10704,12772,15182,18024,21368,25254,29808,35136,41308,48504,56880,66552,77776
%N A102186 The PDO(n) function (Partitions with Designated summands in which all parts are Odd): the sum of products of multiplicities of parts in all partitions of n into odd parts.
%H A102186 Seiichi Manyama, <a href="/A102186/b102186.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H A102186 G. E. Andrews, R. P. Lewis, and J. Lovejoy, <a href="http://dx.doi.org/10.4064/aa105-1-5">Partitions with designated summands</a>, Acta Arith. 105 (2002), no. 1, 51-66.
%H A102186 Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, <a href="https://arxiv.org/abs/2507.10965">Convolutive sequences, I: Through the lens of integer partition functions</a>, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 10, 12.
%H A102186 Nayandeep Deka Baruah and Kanan Kumari Ojah, <a href="https://www.emis.de/journals/INTEGERS/papers/p9/p9.Abstract.html">Partitions with designated summands in which all parts are odd</a>, INTEGERS 15 (2015), #A9.
%H A102186 Shishuo Fu and James Sellers, <a href="https://arxiv.org/abs/2505.21111">A refined view of a curious identity for partitions into odd parts with designated summands</a>, arXiv:2505.21111 [math.CO], 2025.
%F A102186 Euler transform of period 12 sequence [1, 1, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, ...].
%F A102186 a(n) ~ 5^(1/4) * exp(sqrt(5*n)*Pi/3) / (2^(5/2)*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 28 2015
%F A102186 G.f.: Product_{k>=1} (1 + Sum_{j>=1} j * x^(j*(2*k - 1))). - _Ilya Gutkovskiy_, Nov 06 2019
%e A102186 a(8)=22 because in the six partitions of 8 into odd parts, namely, 71,53,5111,3311,311111,11111111, the multiplicities of the parts are (1,1),(1,1),(1,3),(2,2),(1,5),(8) with products 1,1,3,4,5,8, having sum 22.
%p A102186 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A102186        b(n, i-2) +add(b(n-i*j, i-2)*j, j=1..n/i)))
%p A102186     end:
%p A102186 a:= n-> b(n, iquo(1+n,2)*2-1):
%p A102186 seq(a(n), n=0..50);  # _Alois P. Heinz_, Feb 26 2013
%t A102186 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 2] + Sum[b[n - i*j, i - 2]*j, {j, 1, n/i}]]]; a[n_] := b[n, Quotient[1 + n, 2]*2 - 1]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jan 24 2014, after _Alois P. Heinz_ *)
%t A102186 nmax=60; CoefficientList[Series[Product[(1-x^(4*k)) * (1+x^(3*k)) / ((1-x^k) * (1+x^(6*k))), {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Nov 28 2015 *)
%t A102186 Table[Total[l = Tally /@ Select[IntegerPartitions@n, VectorQ[#, OddQ] &];
%t A102186   Table[x = l[[i]]; Product[x[[j, 2]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 47}] (* _Robert Price_, Jun 08 2020 *)
%o A102186 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)*eta(x^6+A)^2/ eta(x+A)/eta(x^3+A)/eta(x^12+A), n))} /* _Michael Somos_, Jul 30 2006 */
%Y A102186 Cf. A077285 (partitions with designated summands).
%K A102186 easy,nonn
%O A102186 0,3
%A A102186 _Vladeta Jovovic_, Feb 16 2005
%E A102186 More terms from _Emeric Deutsch_, Mar 28 2005
%E A102186 Name expanded by _N. J. A. Sloane_, Nov 21 2015