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 A101198 Number of partitions of n with rank 1 (the rank of a partition is the largest part minus the number of parts). 10
 0, 1, 0, 1, 1, 2, 1, 3, 3, 5, 5, 8, 8, 13, 14, 20, 23, 31, 35, 48, 55, 72, 84, 108, 126, 160, 187, 233, 275, 340, 398, 489, 574, 697, 819, 988, 1158, 1390, 1627, 1941, 2271, 2696, 3145, 3721, 4335, 5104, 5938, 6967, 8088, 9462, 10964, 12783 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Column k=1 in the triangle A063995. REFERENCES George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976. LINKS FORMULA G.f. for the number of partitions of n with rank r is Sum((-1)^k*x^(r*k)*(x^((3*k^2+k)/2)-x^((3*k^2-k)/2)), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004 Also Sum(x^(2*n+r+1)*Product((1-x^(2*n+r+1-k))/(1-x^k),k=1..n),n=0..infinity). - Vladeta Jovovic, May 05 2008 EXAMPLE a(6)=2 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively. MAPLE with(combinat): for n from 1 to 35 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=1 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n], n=1..35); MATHEMATICA Table[Count[IntegerPartitions[n], _?(Max[#]-Length[#]==1&)], {n, 60}] (* Harvey P. Dale, Nov 29 2014 *) CROSSREFS Cf. A000041, A063995. Cf. A101198-A101200, A101707-A101709. Sequence in context: A074500 A107237 A070047 * A034394 A058689 A173510 Adjacent sequences:  A101195 A101196 A101197 * A101199 A101200 A101201 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 12 2004 STATUS approved

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Last modified September 24 07:28 EDT 2020. Contains 337317 sequences. (Running on oeis4.)