A092319 -id:A092319 - 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

A092316 Sum of largest parts of all partitions of n into odd distinct parts.

Original entry on oeis.org

1, 0, 3, 3, 5, 5, 7, 12, 14, 16, 18, 27, 29, 33, 42, 55, 59, 65, 78, 95, 110, 118, 137, 167, 188, 200, 236, 274, 303, 330, 376, 435, 485, 522, 591, 677, 741, 803, 903, 1022, 1115, 1210, 1345, 1505, 1650, 1784, 1964, 2201, 2393, 2578, 2843, 3143, 3409, 3685, 4034

Offset: 1

Vladeta Jovovic, Feb 15 2004

			a(13) = 29 because the partitions of 13 into distinct odd parts are [13],[9,3,1] and [7,5,1], with sum of largest terms 13+9+7 = 29.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Seiichi Manyama)
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Cf. A000700, A005895, A067619, A092319, A116857.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1 or i^2 `if`(t>n, 0, b(n-t, i-1)))(2*i-1) ))
    end:
a:= n-> add(`if`(j::odd, j*b(n-j, (j-1)/2), 0), j=1..n):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 19 2022

nmax = 50; Rest[CoefficientList[Series[Sum[(2*k - 1)*x^(2*k - 1) * Product[1 + x^(2*j - 1), {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 28 2016 *)

G.f.: Sum_{n>=1} (2*n-1)*x^(2*n-1)*Product_{k=1..n-1} (1+x^(2*k-1)).

a(n) = 2 * A067619(n) - A000700(n). - Seiichi Manyama, Jan 19 2022

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

A116860 Triangle read by rows: T(n,k) is the number of partitions into distinct odd parts with smallest part k (n>=1, k>=1).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Emeric Deutsch, Feb 27 2006

Row 2 has no terms, row 2n-1 has 2n-1 terms, row 4n has 2n-1 terms, row 4n+2 for n>=1 has 2n-1 terms. Row sums are A000700. T(n,1) = A027349(n). Sum_{k>=1} k*T(n,k) = A092319(n).

Within those rows, T(n, k) = 0 occurs with an odd k iff n is odd and 2*floor((n+3)/6) - 1 <= k < n. - Álvar Ibeas, Aug 03 2020

			T(25,3) = 3 because we have [17,5,3], [15,7,3] and [13,9,3].
Triangle starts:
1;
{};
0,0,1;
1;
0,0,0,0,1;
1;
0,0,0,0,0,0,1;
1,0,1;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A000700, A027349, A092319, A116857.

g:=sum(t^(2*j-1)*x^(2*j-1)*product(1+x^(2*i-1),i=j+1..30),j=1..30): gser:=simplify(series(g,x=0,52)): for n from 1 to 19 do P[n]:=sort(coeff(gser,x^n)) od: d:=proc(n) if n mod 2 = 1 then n elif n=2 then 0 elif n mod 4 = 0 then n/2-1 else n/2-2 fi end: 1; {}; for n from 3 to 19 do seq(coeff(P[n],t^j),j=1..d(n)) od; # yields sequence in triangular form

imax = 20;
s = Sum[t^(2 j - 1)*x^(2 j - 1)*Product[1 + x^(2 i - 1), {i, j + 1, imax}], {j, 1, imax}] + O[x]^imax;
Rest /@ DeleteCases[CoefficientList[#, t]& /@ CoefficientList[s, x], {}] // Flatten (* Jean-François Alcover, May 22 2018 *)

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

For k even, T(n, k) = 0. For k odd, T(n, n) = 1 and, if k < n, T(n, k) = Sum_{i > k} T(n - k, i). - Álvar Ibeas, Aug 03 2020

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

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A092316 Sum of largest parts of all partitions of n into odd distinct parts.

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A116860 Triangle read by rows: T(n,k) is the number of partitions into distinct odd parts with smallest part k (n>=1, k>=1).

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