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A024791 Number of 7's in all partitions of n. 12
0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 23, 32, 45, 61, 84, 112, 151, 199, 263, 342, 446, 574, 739, 943, 1201, 1518, 1917, 2404, 3010, 3749, 4661, 5766, 7122, 8759, 10753, 13153, 16059, 19544, 23743, 28759, 34774, 41938, 50491, 60642, 72718, 87004, 103934, 123908 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

The sums of seven successive terms give A000070. - Omar E. Pol, Jul 12 2012

a(n) is also the difference between the sum of 7th largest and the sum of 8th largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

LINKS

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

FORMULA

a(n) = A181187(n,7) - A181187(n,8). - Omar E. Pol, Oct 25 2012

a(n) ~ exp(Pi*sqrt(2*n/3)) / (14*Pi*sqrt(2*n)) * (1 - 85*Pi/(24*sqrt(6*n)) + (85/48 + 4873*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 05 2016

G.f.: x^7/(1 - x^7) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 06 2017

MAPLE

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

      if n=0 or i=1 then [1, 0]

    else g:= `if`(i>n, [0$2], b(n-i, i));

         b(n, i-1) +g +[0, `if`(i=7, g[1], 0)]

      fi

    end:

a:= n-> b(n, n)[2]:

seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

MATHEMATICA

<< DiscreteMath`Combinatorica`; Table[ Count[ Flatten[ Partitions[n]], 7], {n, 1, 52} ]

Table[Count[Flatten[IntegerPartitions[n]], 7], {n, 55}] (* Harvey P. Dale, Feb 26 2015 *)

b[n_, i_] := b[n, i] = Module[{g}, If[n == 0 || i == 1, {1, 0}, g = If[i > n, {0, 0}, b[n - i, i]]; b[n, i - 1] + g + {0, If[i == 7, g[[1]], 0]}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* Jean-Fran├žois Alcover, Oct 09 2015, after Alois P. Heinz *)

PROG

(PARI) x='x+O('x^50); concat([0, 0, 0, 0, 0, 0], Vec(x^7/(1 - x^7) * prod(k=1, 50, 1/(1 - x^k)))) \\ Indranil Ghosh, Apr 06 2017

CROSSREFS

Cf. A066633, A024786, A024787, A024788, A024789, A024790, A024792, A024793, A024794.

Sequence in context: A239054 A241725 A022480 * A178240 A118084 A232481

Adjacent sequences:  A024788 A024789 A024790 * A024792 A024793 A024794

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified April 25 21:06 EDT 2019. Contains 322461 sequences. (Running on oeis4.)