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A025155 Number of partitions of n into distinct parts >= 10.

%I A025155 #24 Nov 24 2020 16:39:56
%S A025155 1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,6,7,7,9,
%T A025155 9,11,12,14,15,18,19,22,24,27,29,33,36,40,44,49,54,60,66,73,81,89,98,
%U A025155 108,119,130,144,157,173,189,208,227,250,272,299,326,358,389,427,464,508,553
%N A025155 Number of partitions of n into distinct parts >= 10.
%H A025155 Alois P. Heinz, <a href="/A025155/b025155.txt">Table of n, a(n) for n = 0..1000</a>
%F A025155 a(n) = A026830(n+9). - _R. J. Mathar_, Jul 31 2008
%F A025155 G.f.: Product_{j >= 10} (1+x^j). - _R. J. Mathar_, Jul 31 2008
%F A025155 G.f.: Sum_{k>=0} x^(k*(k + 19)/2) / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Nov 24 2020
%e A025155 Say n = 11. Then there is one and only one partition of n into distinct parts each of which is >= 10, namely 11 = 11. Hence a(11) = 1.
%p A025155 b:= proc(n, i) option remember;
%p A025155       `if`(n=0, 1, `if`((i-9)*(i+10)/2<n, 0,
%p A025155        add(b(n-i*j, i-1), j=0..min(1, n/i))))
%p A025155     end:
%p A025155 a:= n-> b(n$2):
%p A025155 seq(a(n), n=0..100);  # _Alois P. Heinz_, Feb 07 2014
%t A025155 dpgt10[n_]:=Module[{prtns=Select[IntegerPartitions[n],Min[#]>9&]}, Count[ prtns,_?(Max[Transpose[Tally[#]][[2]]]==1&)]]; Join[{1},Array[ dpgt10,80]] (* _Harvey P. Dale_, Jun 04 2012 *)
%t A025155 (* also *)
%t A025155 d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 10 &]; Table[d[n], {n, 25}] (* str partitions, parts >= 10 *) Table[Length[d[n]], {n, 40}] (* A025155 for n >= 1 *)
%t A025155 (* _Clark Kimberling_, Mar 07 2014 *)
%t A025155 b[n_, i_] := b[n, i] = If[n==0, 1, If[(i-9)*(i+10)/2<n, 0, Sum[b[n-i*j, i- 1], {j, 0, Min[1, n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 17 2015, after _Alois P. Heinz_ *)
%Y A025155 Cf. A026831, A025147.
%K A025155 nonn
%O A025155 0,22
%A A025155 _Clark Kimberling_