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 A024786 Number of 2's in all partitions of n. 42
 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, 295, 389, 526, 686, 911, 1176, 1538, 1968, 2540, 3223, 4115, 5181, 6551, 8191, 10269, 12756, 15873, 19598, 24222, 29741, 36532, 44624, 54509, 66261, 80524, 97446, 117862, 142029, 171036, 205290, 246211 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Also number of partitions of n-1 with a distinguished part different from all the others. [Comment corrected by Emeric Deutsch, Aug 13 2008] In general the number of times that j appears in the partitions of n equals Sum_{k=3} 1/(1-x^j) from Riordan reference second term, last eq. a(n) = A006128(n-1) - A194452(n-1). - Omar E. Pol, Nov 20 2011 a(n) = A181187(n,2) - A181187(n,3). - Omar E. Pol, Oct 25 2012 a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2) * Pi * sqrt(n)) * (1 - 25*Pi/(24*sqrt(6*n)) + (25/48 + 433*Pi^2/6912)/n). - Vaclav Kotesovec, Mar 07 2016, extended Nov 05 2016 a(n) = Sum_{k} k * A116595(n-1,k). - Emeric Deutsch, Sep 12 2016 G.f.: x^2/((1 - x)*(1 - x^2)) * Sum_{n >= 0} x^(2*n)/( Product_{k = 1..n} 1 - x^k ); that is, convolution of A004526 (partitions into 2 parts, or, modulo offset differences, partitions into parts <= 2) and A002865 (partitions into parts >= 2). - Peter Bala, Jan 17 2021 EXAMPLE From Omar E. Pol, Oct 25 2012: (Start) For n = 7 we have: -------------------------------------- . Number Partitions of 7 of 2's -------------------------------------- 7 .............................. 0 4 + 3 .......................... 0 5 + 2 .......................... 1 3 + 2 + 2 ...................... 2 6 + 1 .......................... 0 3 + 3 + 1 ...................... 0 4 + 2 + 1 ...................... 1 2 + 2 + 2 + 1 .................. 3 5 + 1 + 1 ...................... 0 3 + 2 + 1 + 1 .................. 1 4 + 1 + 1 + 1 .................. 0 2 + 2 + 1 + 1 + 1 .............. 2 3 + 1 + 1 + 1 + 1 .............. 0 2 + 1 + 1 + 1 + 1 + 1 .......... 1 1 + 1 + 1 + 1 + 1 + 1 + 1 ...... 0 ------------------------------------ . 24 - 13 = 11 . The difference between the sum of the second column and the sum of the third column of the set of partitions of 7 is 24 - 13 = 11 and equals the number of 2's in all partitions of 7, so a(7) = 11. (End) MAPLE b:= proc(n, i) option remember; local f, g; if n=0 or i=1 then [1, 0] else f:= b(n, i-1); g:= `if`(i>n, [0\$2], b(n-i, i)); [f[1]+g[1], f[2]+g[2]+`if`(i=2, g[1], 0)] fi end: a:= n-> b(n, n)[2]: seq(a(n), n=1..50); # Alois P. Heinz, May 18 2012 MATHEMATICA Table[ Count[ Flatten[ IntegerPartitions[n]], 2], {n, 1, 50} ] (* Second program: *) b[n_, i_] := b[n, i] = Module[{f, g}, If[n==0 || i==1, {1, 0}, f = b[n, i - 1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i == 2, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 22 2015, after Alois P. Heinz *) Join[{0}, (1/((1 - x^2) QPochhammer[x]) + O[x]^50)[[3]]] (* Vladimir Reshetnikov, Nov 22 2016 *) Table[Sum[(1 + (-1)^k)/2 * PartitionsP[n-k], {k, 2, n}], {n, 1, 50}] (* Vaclav Kotesovec, Aug 27 2017 *) PROG (Python) from sympy import npartitions def A024786(n): return sum(npartitions(n-(k<<1)) for k in range(1, (n>>1)+1)) # Chai Wah Wu, Oct 25 2023 CROSSREFS Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794, A173238. Column 2 of A060244. First differences of A000097. Sequence in context: A212548 A212549 A212550 * A299069 A097497 A332681 Adjacent sequences: A024783 A024784 A024785 * A024787 A024788 A024789 KEYWORD nonn AUTHOR Clark Kimberling STATUS approved

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Last modified May 18 08:45 EDT 2024. Contains 372618 sequences. (Running on oeis4.)