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A024787 Number of 3's in all partitions of n.

%I A024787 #64 Mar 28 2025 12:49:38
%S A024787 0,0,1,1,2,4,6,9,15,21,31,45,63,87,122,164,222,298,395,519,683,885,
%T A024787 1146,1475,1887,2401,3050,3845,4837,6060,7563,9402,11664,14405,17751,
%U A024787 21807,26715,32634,39784,48352,58649,70969,85690,103232,124143,148951,178407,213277,254509
%N A024787 Number of 3's in all partitions of n.
%C A024787 Starting with the first 1 = row sums of triangle A173239. - _Gary W. Adamson_, Feb 13 2010
%C A024787 The sums of three successive terms give A000070. - _Omar E. Pol_, Jul 12 2012
%C A024787 a(n) is also the difference between the sum of 3rd largest and the sum of 4th largest elements in all partitions of n. - _Omar E. Pol_, Oct 25 2012
%H A024787 Alois P. Heinz, <a href="/A024787/b024787.txt">Table of n, a(n) for n = 1..5000</a>
%H A024787 David Benson, Radha Kessar, and Markus Linckelmann, <a href="https://arxiv.org/abs/2204.09970">Hochschild cohomology of symmetric groups in low degrees</a>, arXiv:2204.09970 [math.GR], 2022.
%H A024787 Philip Cuthbertson, <a href="https://math.colgate.edu/~integers/z28/z28.pdf">Fixed hooks in arbitrary columns of partitions</a>, Integers (2025) Vol. 25, Art. No. A28. See p. 3.
%H A024787 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%F A024787 a(n) = A181187(n,3) - A181187(n,4). - _Omar E. Pol_, Oct 25 2012
%F A024787 a(n) = Sum_{k=1..floor(n/3)} A263232(n,k). - _Alois P. Heinz_, Nov 01 2015
%F A024787 a(n) ~ exp(Pi*sqrt(2*n/3)) / (6*Pi*sqrt(2*n)) * (1 - 37*Pi/(24*sqrt(6*n)) + (37/48 + 937*Pi^2/6912)/n). - _Vaclav Kotesovec_, Nov 05 2016
%F A024787 G.f.: x^2/((1 - x^3)*(x)_inf), where (q)_inf is the q-Pochhammer symbol (the Euler function). - _Vladimir Reshetnikov_, Nov 22 2016
%F A024787 G.f.: x^3/((1 - x)*(1 - x^2)*(1 - x^3)) * Sum_{n >= 0} x^(3*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A069905 (partitions into 3 parts, or, modulo offset differences, partitions into parts <= 3) and A008483 (partitions into parts >= 3). - _Peter Bala_, Jan 17 2021
%e A024787 From _Omar E. Pol_, Oct 25 2012: (Start)
%e A024787 For n = 7 we have:
%e A024787 --------------------------------------
%e A024787 .                             Number
%e A024787 Partitions of 7               of 3's
%e A024787 --------------------------------------
%e A024787 7 .............................. 0
%e A024787 4 + 3 .......................... 1
%e A024787 5 + 2 .......................... 0
%e A024787 3 + 2 + 2 ...................... 1
%e A024787 6 + 1 .......................... 0
%e A024787 3 + 3 + 1 ...................... 2
%e A024787 4 + 2 + 1 ...................... 0
%e A024787 2 + 2 + 2 + 1 .................. 0
%e A024787 5 + 1 + 1 ...................... 0
%e A024787 3 + 2 + 1 + 1 .................. 1
%e A024787 4 + 1 + 1 + 1 .................. 0
%e A024787 2 + 2 + 1 + 1 + 1 .............. 0
%e A024787 3 + 1 + 1 + 1 + 1 .............. 1
%e A024787 2 + 1 + 1 + 1 + 1 + 1 .......... 0
%e A024787 1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0
%e A024787 ------------------------------------
%e A024787 .      13 - 7 =                  6
%e A024787 The difference between the sum of the third column and the sum of the fourth column of the set of partitions of 7 is 13 - 7 = 6 and equals the number of 3's in all partitions of 7, so a(7) = 6.
%e A024787 (End)
%p A024787 b:= proc(n, i) option remember; local g;
%p A024787       if n=0 or i=1 then [1, 0]
%p A024787     else g:= `if`(i>n, [0$2], b(n-i, i));
%p A024787          b(n, i-1) +g +[0, `if`(i=3, g[1], 0)]
%p A024787       fi
%p A024787     end:
%p A024787 a:= n-> b(n, n)[2]:
%p A024787 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 27 2012
%t A024787 Table[ Count[ Flatten[ IntegerPartitions[n]], 3], {n, 1, 50} ]
%t A024787 b[n_, i_] := b[n, i] = Module[{g}, If[n==0 || i==1, {1, 0}, g = If[i>n, {0, 0}, b[n-i, i]]; b[n, i-1] + g + {0, If[i==3, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Oct 09 2015, after _Alois P. Heinz_ *)
%t A024787 Join[{0, 0}, (1/((1 - x^3) QPochhammer[x]) + O[x]^50)[[3]]] (* _Vladimir Reshetnikov_, Nov 22 2016 *)
%Y A024787 Cf. A066633, A024786, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A263232.
%Y A024787 Cf. A173239. - _Gary W. Adamson_, Feb 13 2010
%K A024787 nonn,easy
%O A024787 1,5
%A A024787 _Clark Kimberling_