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 A025147 Number of partitions of n into distinct parts >= 2. 77
 1, 0, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 17, 21, 25, 29, 35, 41, 48, 56, 66, 76, 89, 103, 119, 137, 159, 181, 209, 239, 273, 312, 356, 404, 460, 522, 591, 669, 757, 853, 963, 1085, 1219, 1371, 1539, 1725, 1933, 2164, 2418, 2702, 3016, 3362, 3746, 4171, 4637, 5155 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS From R. J. Mathar, Jul 31 2008: (Start) These "partitions of n into distinct parts >= k" and "partitions of n into distinct parts, the least being k-1" come in pairs of similar, almost shifted but not identical, sequences: A025147, A096765 (k=2) A025148, A096749 (k=3) A025149, A026824 (k=4) A025150, A026825 (k=5) A025151, A026826 (k=6) A025152, A026827 (k=7) A025153, A026828 (k=8) A025154, A026829 (k=9) A025155, A026830 (k=10) A096740, A026831 (k=11) The distinction in the definitions is that "distinct parts >= k" sets a lower bound to all parts, whereas "the least being ..." means that the lower limit must be attained by one of the parts. (End) From N. J. A. Sloane, Sep 28 2008: (Start) Generating functions and Maple programs for the sequences in the first and second columns of the above list are respectively: For A025147, A025148, etc.: f:=proc(k) product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); end; For A096765, A096749, etc.: g:=proc(k) x^(k-1)*product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); end; (End) Also number of partitions of n+1 into distinct parts, the least being 1. Number of different sums from 1+[1,3]+[1,4]+...+[1,n]. - Jon Perry, Jan 01 2004 Also number of partitions of n such that if k is the largest part, then all parts from 1 to k occur, k occurring at least twice. Example: a(7)=3 because we have [2,2,2,1],[2,2,1,1,1] and [1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 09 2006 Also number of partitions of n+1 such that if k is the largest part, then all parts from 1 to k occur, k occurring exactly once. Example: a(7)=3 because we have [3,2,2,1],[3,2,1,1,1] and [2,1,1,1,1,1,1] (there is a simple bijection with the partitions defined before). - Emeric Deutsch, Apr 09 2006 Also number of partitions of n+1 into distinct parts where the number of parts is itself a part. - Reinhard Zumkeller, Nov 04 2007 Partial sums give A038348 (observed by Jonathan Vos Post, proved by several correspondents). Trivially, number of partitions of n into distinct parts (as ascending lists) such that the first part is not 1, the second not 2, the third not 3, etc., see example. - Joerg Arndt, Jun 10 2013 Convolution with A033999 gives A270144 (apart from the offset). - R. J. Mathar, Jun 18 2016 REFERENCES Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997.  MathEduc Database (Zentralblatt MATH, 1997c.01891). Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II,   Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.  Zentralblatt MATH, Zbl 1071.05501. LINKS Reinhard Zumkeller and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 0..100 from Reinhard Zumkeller) Rebekah Ann Gilbert, A Fine Rediscovery, 2014. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 798 Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q_1(n), see Corollary 3.7. FORMULA G.f.: Product_{k=2..infinity} (1+x^k). a(n) = A000009(n)-a(n-1) = Sum_{0<=k<=n} (-1)^k*A000009(n-k). - Henry Bottomley, May 09 2002 a(n)=t(n, 1), where t(n, k)=1+Sum_{i>j>k and i+j=n} t(i, j), 2<=k<=n. - Reinhard Zumkeller, Jan 01 2003 G.f.: 1 + Sum_{k=1..infinity} (x^(k*(k+3)/2) / Product_{j=1..k} (1-x^j)). - Emeric Deutsch, Apr 09 2006 The previous g.f. is a special case of the g.f. for partitions into distinct parts >= L, Sum_{n>=0} ( x^(n*(n+2*L-1)/2) / Product_{k=1..n} (1-x^k) ). - Joerg Arndt, Mar 24 2011 G.f.: Sum_{n>=1} ( x^(n*(n+1)/2-1) / Product_{k=1..n-1} (1-x^k) ), a special case of the g.f. for partitions into distinct parts >= L, Sum_{n>=L-1} ( x^(n*(n+1)/2-L*(L-1)/2) / Product_{k=1..n-(L-1)} (1-x^k) ). - Joerg Arndt, Mar 27 2011 a(n) = Sum_{10. - Reinhard Zumkeller, Nov 04 2007 a(n) = A096765(n+1). - R. J. Mathar, Jul 31 2008 From Vaclav Kotesovec, Aug 16 2015: (Start) a(n) ~ 1/2 * A000009(n). a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). (End) EXAMPLE a(7) = 3, from {{3, 4}, {2, 5}, {7}} From Joerg Arndt, Jun 10 2013: (Start) There are a(17) = 21 partitions of 17 into distinct parts >=2: 01:  [ 2 3 4 8 ] 02:  [ 2 3 5 7 ] 03:  [ 2 3 12 ] 04:  [ 2 4 5 6 ] 05:  [ 2 4 11 ] 06:  [ 2 5 10 ] 07:  [ 2 6 9 ] 08:  [ 2 7 8 ] 09:  [ 2 15 ] 10:  [ 3 4 10 ] 11:  [ 3 5 9 ] 12:  [ 3 6 8 ] 13:  [ 3 14 ] 14:  [ 4 5 8 ] 15:  [ 4 6 7 ] 16:  [ 4 13 ] 17:  [ 5 12 ] 18:  [ 6 11 ] 19:  [ 7 10 ] 20:  [ 8 9 ] 21:  [ 17 ] (End) MAPLE g:=product(1+x^j, j=2..65): gser:=series(g, x=0, 62): seq(coeff(gser, x, n), n=0..57); # Emeric Deutsch, Apr 09 2006 with(combstruct):ZL := {L = PowerSet(Sequence(Z, card>=2)) }, unlabeled:seq(count([L, ZL], size=i), i=0..57); # Zerinvary Lajos, Mar 09 2007 MATHEMATICA CoefficientList[Series[Product[1+q^n, {n, 2, 60}], {q, 0, 60}], q] FoldList[ PartitionsQ[ #2+1 ]-#1&, 0, Range[ 64 ] ] (* also *) d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 2 &]; Table[d[n], {n, 12}] (* strict partitions, parts >= 2 *) Table[Length[d[n]], {n, 40}] (* A025147 for n >= 1 *) (* Clark Kimberling, Mar 07 2014 *) p[_, 0] = 1; p[k_, m_] := p[k, m] = If[m < k, 0, p[k+1, m-k] + p[k+1, m]]; Table[p[2, m], {m, 0, 59}] (* Jean-François Alcover, Apr 17 2014, after Reinhard Zumkeller *) PROG (Haskell) a025147 = p 2 where    p _ 0 = 1    p k m = if m < k then 0 else p (k + 1) (m - k) + p (k + 1) m -- Reinhard Zumkeller, Dec 28 2011 (PARI) a(n)=if(n, my(v=partitions(n)); sum(i=1, #v, v[i]>1&&v[i]==vecsort(v[i], , 8)), 1) \\ Charles R Greathouse IV, Nov 20 2012 CROSSREFS Cf. A015744, A015745, A015746, A015750, A015753, A015754, A015755, A002865. Sequence in context: A238217 A185226 A096765 * A032230 A238789 A126793 Adjacent sequences:  A025144 A025145 A025146 * A025148 A025149 A025150 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Corrected and extended by Dean Hickerson, Oct 10 2001 STATUS approved

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Last modified May 10 02:36 EDT 2021. Contains 343747 sequences. (Running on oeis4.)