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 A238869 Number of partitions of n where the difference between consecutive parts is at most 9. 10
 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 131, 170, 221, 283, 364, 461, 586, 737, 926, 1154, 1439, 1779, 2199, 2703, 3317, 4051, 4942, 6001, 7278, 8796, 10610, 12760, 15323, 18344, 21928, 26148, 31127, 36971, 43848, 51890, 61321, 72327, 85183, 100149, 117588, 137827, 161343, 188583, 220139, 256607, 298761, 347360 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most nine times (by taking conjugates). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(10*i))/(1-q^i) ) ). a(n) = Sum_{k=0..9} A238353(n,k). - Alois P. Heinz, Mar 09 2014 MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(b(n-i*j, i-1), j=0..min(9, n/i))))     end: g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(b(n-i*j, i-1), j=1..n/i)))     end: a:= n-> add(g(n, k), k=0..n): seq(a(n), n=0..60);  # Alois P. Heinz, Mar 09 2014 MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 0, Min[9, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *) PROG (PARI) N=66;  q = 'q + O('q^N); Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1, k-1, (1-q^(10*i))/(1-q^i) ) ) ) CROSSREFS Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1),  A034296 (d=1), A224956 (d=2), A238863 (d=3), A238864 (d=4), A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), this sequence, A000041 (d --> infinity). Sequence in context: A218511 A008640 A008634 * A326333 A036011 A325856 Adjacent sequences:  A238866 A238867 A238868 * A238870 A238871 A238872 KEYWORD nonn AUTHOR Joerg Arndt, Mar 08 2014 STATUS approved

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Last modified May 7 19:46 EDT 2021. Contains 343652 sequences. (Running on oeis4.)