A092316 -id:A092316 - cp's OEIS frontend

A092265 Sum of smallest parts of all partitions of n into distinct parts.

1, 2, 4, 5, 8, 10, 14, 16, 23, 26, 34, 40, 50, 58, 74, 83, 102, 120, 142, 164, 198, 226, 266, 308, 359, 412, 482, 548, 634, 730, 834, 950, 1094, 1240, 1416, 1609, 1826, 2068, 2350, 2648, 2994, 3382, 3806, 4280, 4826, 5408, 6070, 6806, 7619, 8522, 9534, 10632

Offset: 1

Vladeta Jovovic, Feb 14 2004

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A046746, A005895, A006128, A092269, A092319, A092316, A034296, A237665.

Cf. A026832, A336902, A336903.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n,i+1)+b(n-i, i+1)))
    end:
a:= n-> add(j*b(n-j, j+1), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i + 1] + b[n - i, i + 1]]]; a[n_] := Sum[j*b[n - j, j + 1], {j, 1, n}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *)

G.f.: Sum_{n >= 1} (-1 + Product_{k >= n} 1 + x^k).
G.f.: Sum_{n >= 1} n*x^n*Product_{k >= n+1} (1 + x^k). - Joerg Arndt, Jan 29 2011
G.f.: Sum_{k >= 1} x^(k*(k+1)/2)/(1 - x^k)/Product_{i = 1..k} (1 - x^i). - Vladeta Jovovic, Aug 10 2004
Conjecture: a(n) = A034296(n) + A237665(n+1). - George Beck, May 06 2017
a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 20 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A092319 Sum of smallest parts of all partitions of n into odd distinct parts.

Original entry on oeis.org

1, 0, 3, 1, 5, 1, 7, 4, 10, 4, 12, 9, 15, 9, 20, 17, 23, 17, 28, 27, 36, 28, 41, 43, 50, 44, 62, 62, 71, 66, 84, 91, 103, 96, 119, 127, 139, 137, 167, 178, 191, 192, 223, 241, 266, 264, 302, 331, 351, 360, 411, 439, 469, 485, 542, 587, 628, 646, 714, 773, 819, 854, 945

Offset: 1

Vladeta Jovovic, Feb 15 2004

a(n) = Sum_{k>=0} A116860(n,k). - Emeric Deutsch, Feb 27 2006

			a(13)=15 because the partitions of 13 into distinct odd parts are [13],[9,3,1] and [7,5,1], with sum of the smallest terms 13+1+1=15.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A092316.

Cf. A116860.

f:=sum((2*n-1)*x^(2*n-1)*product(1+x^(2*k+1),k=n..40),n=1..40): fser:=simplify(series(f,x=0,66)): seq(coeff(fser,x^n),n=1..63); # Emeric Deutsch, Feb 27 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+2)+b(n-i, i+2)))
    end:
a:= n-> add(`if`(j::odd, j*b(n-j, j+2), 0), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

nmax = 60; Rest[CoefficientList[Series[Sum[(2*k - 1)*x^(2*k - 1) * Product[1 + x^(2*j + 1), {j, k, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 28 2016 *)

G.f.: Sum((2*n-1)*x^(2*n-1)*Product(1+x^(2*k+1), k = n .. infinity), n = 1 .. infinity).

a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/6)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, May 20 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A116857 Triangle read by rows: T(n,k) is the number of partitions of n into distinct odd parts, the largest of which is k (n>=1, k>=1).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Emeric Deutsch, Feb 24 2006

Both rows 2n-1 and 2n have 2n-1 terms each. Row sums yield A000700. T(n,2k)=0 Sum(k*T(n,k),k>=1) = A092316(n).

			T(20,11)=2 because we have [11,9] and [11,5,3,1].
T(30,17)=3 because we have [17,13],[17,9,3,1] and [17,7,5,1].
Triangle starts:
  1;
  0;
  0,0,1;
  0,0,1;
  0,0,0,0,1;
  0,0,0,0,1;
  0,0,0,0,0,0,1;
  0,0,0,0,1,0,1;
  ...

Alois P. Heinz, Rows n = 1..350, flattened

Cf. A000700, A092316.

g:=sum(t^(2*j-1)*x^(2*j-1)*product(1+x^(2*i-1),i=1..j-1),j=1..30): gser:=simplify(series(g,x=0,22)): for n from 1 to 20 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 20 do seq(coeff(P[n],t^j),j=1..2*ceil(n/2)-1) od; # yields sequence in triangular form

G.f.: sum(t^(2j-1)*x^(2j-1)*product(1+x^(2i-1), i=1..j-1), j=1..infinity).

cp's OEIS Frontend

A092265 Sum of smallest parts of all partitions of n into distinct parts.

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A092319 Sum of smallest parts of all partitions of n into odd distinct parts.

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A116857 Triangle read by rows: T(n,k) is the number of partitions of n into distinct odd parts, the largest of which is k (n>=1, k>=1).

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