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A207033 Total number of parts >= 3 in all partitions of n. 2
0, 0, 1, 2, 4, 8, 13, 22, 35, 54, 80, 121, 172, 247, 347, 484, 661, 906, 1215, 1632, 2162, 2855, 3730, 4871, 6290, 8111, 10381, 13252, 16802, 21269, 26750, 33583, 41948, 52277, 64862, 80326, 99055, 121922, 149541, 183052, 223350, 272038, 330343, 400450, 484154 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: Sum_{k>=1} x^(3*k)/(1 - x^k) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

EXAMPLE

a(4) = 2, because 2 parts have size >= 3 in all partitions of 4: [1,1,1,1], [1,1,2], [2,2], [1,3], [4].

MAPLE

b:= proc(n, i) option remember; local f, g;

      if n=0 then [1, 0]

    elif i<1 then [0, 0]

    elif i>n then b(n, i-1)

    else f:= b(n, i-1); g:= 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, Feb 19 2012

MATHEMATICA

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i < 1, {0, 0}, i > n, b[n, i - 1], True, f = b[n, i - 1]; g = b[n - i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i > 2, g[[1]], 0]}]];

a[n_] := b[n, n][[2]];

Array[a, 50] (* Jean-Fran├žois Alcover, Nov 12 2020, after Alois P. Heinz *)

CROSSREFS

Column 3 of A181187.

Cf. A006128, A096541, A206563.

Sequence in context: A351620 A078157 A144119 * A291553 A330153 A244985

Adjacent sequences:  A207030 A207031 A207032 * A207034 A207035 A207036

KEYWORD

nonn

AUTHOR

Omar E. Pol, Feb 18 2012

EXTENSIONS

More terms from Alois P. Heinz, Feb 18 2012

STATUS

approved

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Last modified May 28 22:08 EDT 2022. Contains 354122 sequences. (Running on oeis4.)