A182977 -id:A182977 - cp's OEIS frontend

A220477 Total number of parts in all partitions of n with at least one distinct part.

0, 0, 2, 5, 14, 23, 46, 71, 115, 174, 263, 371, 542, 756, 1044, 1432, 1947, 2605, 3478, 4588, 6020, 7863, 10182, 13114, 16820, 21480, 27254, 34489, 43423, 54491, 68103, 84864, 105318, 130408, 160828, 197923, 242774, 297141, 362531, 441456, 536062, 649695

Offset: 1

Omar E. Pol, Jan 16 2013

Also total number of parts in all partitions of n minus the sum of divisors of n. Also sum of largest parts of all partitions of n minus the sum of divisors of n.

			For n = 6
-----------------------------------------------------
Partitions of 6            Value
-----------------------------------------------------
6 .......................... 0  (all parts are equal)
5 + 1 ...................... 2
4 + 2 ...................... 2
4 + 1 + 1 .................. 3
3 + 3 ...................... 0  (all parts are equal)
3 + 2 + 1 .................. 3
3 + 1 + 1 + 1 .............. 4
2 + 2 + 2 .................. 0  (all parts are equal)
2 + 2 + 1 + 1 .............. 4
2 + 1 + 1 + 1 + 1 .......... 5
1 + 1 + 1 + 1 + 1 + 1 ...... 0  (all parts are equal)
-----------------------------------------------------
The sum of the values is    23
On the other hand the total number of parts of the partitions of 6 is A006128(6) = 35 and the sum of divisor of 6 is 1 + 2 + 3 + 6 = sigma(6) = A000203(6) = 12 equals the total number of parts of the partitions of 6 into equal parts. So a(6) = 35 - 12 = 23.

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

Cf. A000005, A000203, A000041, A006128, A066186, A182629, A182977, A182978.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f, g:= b(n, i-1), `if`(i>n, [0$2], b(n-i, i));
         [f[1]+g[1], f[2]+g[2] +g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -numtheory[sigma](n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 17 2013

a[n_] := Sum[DivisorSigma[0, k]*PartitionsP[n-k], {k, 1, n}] - DivisorSigma[1, n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 22 2015 *)

a(n) = A006128(n) - A000203(n).

G.f.: Q(0)/(1-x), where Q(k)= 1 - prod(n=1..k+1, (1-x^n))/( 1 - (x^(k+1)) - x*(1- (x^(k+1)))^2*(k+2)/( x*(1- (x^(k+1)))*(k+2) - (k+1)*(1 - (x^(k+2)))*prod(n=1..k+1, (1-x^n) )/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 16 2013

A182978 Total number of parts that are the smallest part or the largest part in all partitions of n.

Original entry on oeis.org

1, 3, 6, 12, 20, 34, 52, 80, 116, 170, 236, 333, 453, 621, 825, 1111, 1455, 1923, 2487, 3239, 4149, 5342, 6770, 8625, 10852, 13698, 17107, 21413, 26567, 33019, 40721, 50270, 61663, 75665, 92318, 112686, 136849, 166173, 200923, 242836

Offset: 1

Omar E. Pol, Jul 17 2011

nonn

			For n = 6 the partitions of 6 are
6
5 + 1
4 + 2
4 + 1 + 1
3 + 3
3 + (2) + 1 .... the "2" is the part that does not count.
3 + 1 + 1 + 1
2 + 2 + 2
2 + 2 + 1 + 1
2 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1
The total number of parts in all partitions of 6 is equal to 35. All parts are the smallest part or the largest part, except the "2" in the partition (3 + 2 + 1), so a(6) = 35 - 1 = 34.

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

Cf. A006128, A046746, A092269, A116686, A182977, A182984.

l:= proc(n, i) option remember; `if`(n=i, n, 0)+
      `if`(i<1, 0, l(n, i-1) +`if`(n l(n, n) +s(n, n) -numtheory[sigma](n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 17 2013

l[n_, i_] := l[n, i] = If[n==i, n, 0] + If[i<1, 0, l[n, i-1] + If[nJean-François Alcover, Nov 03 2015, after Alois P. Heinz *)

a(n) = A006128(n) - A182977(n).

a(12) corrected and more terms a(13)-a(40) from David Scambler, Jul 18 2011

A265249 Triangle read by rows: T(n,k) is the number of partitions of n having k parts strictly between the smallest and the largest part (n>=1, k>=0).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 1, 13, 2, 17, 4, 1, 20, 8, 2, 26, 11, 4, 1, 29, 17, 8, 2, 35, 24, 13, 4, 1, 39, 33, 19, 8, 2, 48, 39, 30, 13, 4, 1, 48, 56, 41, 21, 8, 2, 60, 64, 57, 32, 13, 4, 1, 61, 83, 75, 47, 21, 8, 2, 74, 94, 100, 65, 34, 13, 4, 1

Offset: 1

Emeric Deutsch, Dec 25 2015

Number of entries in row n is floor((n-4)/2) (n>=4).
Sum of entries of row n = A000041(n) = number of partitions of n.
T(n,0) = A265250(n).
Sum(k*T(n,k), k>=0) = A182977(n).

			T(8,2) = 1 because among the 22 partitions of 8 only [3,2,2,1] has 2 parts strictly between the smallest and the largest part.
Triangle starts:
1;
2;
3;
5;
7;
10, 1;
13, 2;

CF. A000041, A182977, A265250.

g := add(x^i/(1-x^i), i=1..80)+add(add(x^(i+j)/((1-x^i)*(1-x^j)*mul(1-t*x^k, k=i+1..j-1)),j=i+1..80),i=1..80): gser := simplify(series(g,x=0,23)): for n to 22 do P[n]:= sort(coeff(gser,x,n)) end do: for n to 22 do seq(coeff(P[n],t,k), k=0..degree(P[n])) end do; # yields sequence in triangular form

G.f.: G(t,x) = Sum_{i>=1} x^i/(1-x^i) + Sum_{i>=1} Sum_{j>=i+1} x^(i+j)/(1-x^i)/(1-x^j)/Product_{k=i+1..j-1} (1-tx^k).

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A220477 Total number of parts in all partitions of n with at least one distinct part.

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A182978 Total number of parts that are the smallest part or the largest part in all partitions of n.

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A265249 Triangle read by rows: T(n,k) is the number of partitions of n having k parts strictly between the smallest and the largest part (n>=1, k>=0).

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