A024791 Number of 7's in all partitions of n.

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

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

David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.

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-François 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: A022480 A308927 A366850 * A178240 A118084 A232481

Adjacent sequences: A024788 A024789 A024790 * A024792 A024793 A024794

Keyword

nonn

Author

Clark Kimberling

Status

approved