A087897 -id:A087897 - cp's OEIS frontend

A362829 Positions in lexicographic order of odd partitions of sufficiently large numbers.

1, 3, 7, 10, 15, 20, 27, 30, 39, 41, 51, 56, 69, 72, 75, 93, 95, 101, 123, 128, 132, 134, 160, 163, 166, 172, 176, 212, 214, 220, 227, 229, 273, 278, 282, 284, 291, 297, 353, 356, 359, 365, 369, 379, 382, 384, 453, 455, 461, 468, 470, 481, 483, 490, 579, 584

Offset: 1

Richard Peterson, Aug 01 2023

nonn

a(n) is the position in lexicographic order of the n-th odd partition of a sufficiently large number k. As long as the number k whose partitions we are examining is large enough, a(n) will exist and won't change for different k. The number of partitions of an odd number, for example, 101 for k=13, will always appear in the sequence, since 13 is the 101st partition in lexicographic order.

Equivalently, positions of partitions with all parts odd among all partitions with no parts of size 1, ordered first by sum, then lexicographically (with the parts in nondecreasing order); or positions of partitions with all parts even among all partitions ordered first by the number of parts plus the sum of the parts, then lexicographically. - Pontus von Brömssen, Sep 14 2023

			a(1)=1 because 1+1+...+1 (k times) is the first partition in lexicographic order of any positive integer k, and it is odd.
a(2)=3 because 1+1+...+1(k-3 times)+3=k is the third partition of k lexicographically and it is odd.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A000009, A087897, A000041.

More terms from Pontus von Brömssen, Sep 14 2023

A115604 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts in which the smallest part occurs k times (1<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 2, 1, 0, 1, 0, 0, 1, 3, 2, 1, 1, 1, 0, 1, 0, 0, 1, 3, 3, 1, 1, 1, 1, 0, 1, 0, 0, 1, 4, 2, 2, 2, 1, 1, 1, 0, 1, 0, 0, 1, 5, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 5, 4, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1

Offset: 1

Emeric Deutsch, Mar 13 2006

Row sums yield A000009. T(n,1)=A087897(n+2). Sum(k*T(n,k),k=1..n)=A092268(n).

			T(14,2)=4 because we have [9,3,1,1],[7,7],[7,5,1,1] and [3,3,3,3,1,1].
Triangle starts:
1;
0,1;
1,0,1;
1,0,0,1;
1,1,0,0,1;
1,1,1,0,0,1;
2,1,0,1,0,0,1;

Cf. A000009, A087897, A092268, A117408.

g:=sum(t*x^(2*k-1)/(1-t*x^(2*k-1))/product(1-x^(2*i-1),i=k+1..40),k=1..40): gser:=simplify(series(g,x=0,55)): for n from 1 to 15 do P[n]:=expand(coeff(gser,x^n)) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form

G.f.=G(t,x)=sum(tx^(2k-1)/[(1-tx^(2k-1))product(1-x^(2i-1), i=k+1..infinity)], k=1..infinity).

A116663 Triangle read by rows: T(n,k) = number of partitions of n into odd parts and having exactly k parts equal to 1 (n>=0, 1<=k<=n).

Original entry on oeis.org

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

Offset: 0

Emeric Deutsch, Feb 22 2006

Row sums yield A000009. T(n,0)=A087897(n). Column k has g.f.=x^k/Product(1-x^(2j-1), j=2..infinity) (all columns are basically identical). Sum(k*T(n,k),k=0..n)=A036469(n).

			T(10,1)=2 because the only partitions of 10 into odd parts and having exactly 1 part equal to 1 are [9,1] and [3,3,3,1].
Triangle starts:
1;
0,1;
0,0,1;
1,0,0,1;
0,1,0,0,1;

Cf. A000009, A087897, A036469.

g:=1/(1-t*x)/product(1-x^(2*j-1),j=2..30): gser:=simplify(series(g,x=0,18)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.=1/[(1-tx)*Product(1-x^(2j-1), j=2..infinity)].

A303902 Expansion of (1 - x^2)*Product_{k>=2} (1 + x^k)^k.

Original entry on oeis.org

1, 0, 1, 3, 3, 8, 12, 21, 34, 59, 93, 150, 242, 377, 595, 922, 1419, 2171, 3310, 4988, 7507, 11218, 16674, 24676, 36353, 53295, 77828, 113209, 163989, 236736, 340517, 488108, 697407, 993350, 1410455, 1996968, 2819280, 3969260, 5573541, 7806141, 10905640, 15199138, 21133212

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

First differences of A026007.

Index entries for sequences related to partitions

Cf. A002865, A026007, A087897, A191659, A298598, A302832.

nmax = 42; CoefficientList[Series[(1 - x^2) Product[(1 + x^k)^k, {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[(1 - x) Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]

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

a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * Zeta(3)^(1/2) / (2^(13/12) * sqrt(Pi) * n). - Vaclav Kotesovec, May 04 2018

A331980 Number of compositions (ordered partitions) of n into distinct odd parts > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 4, 1, 4, 7, 6, 7, 6, 13, 8, 19, 8, 25, 34, 31, 34, 43, 60, 49, 84, 61, 134, 73, 158, 205, 232, 217, 280, 355, 378, 487, 450, 745, 572, 1003, 668, 1381, 1558, 1759, 1678, 2383, 2592, 3001, 3480, 3865, 5162, 4729, 6794, 5953, 9964

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

			a(12) = 4 because we have [9, 3], [7, 5], [5, 7] and [3, 9].

Index entries for sequences related to compositions

Cf. A000931, A027349, A032020, A032021, A032022, A087897.

nmax = 60; CoefficientList[Series[Sum[k! x^(k (k + 2))/Product[(1 - x^(2 j)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(k + 2)) / Product_{j=1..k} (1 - x^(2*j)).

A334377 Irregular triangle read by rows: T(n,k) is the number of partitions of k into distinct parts p such that 2 <= p <= n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 2, 3, 2, 4, 3, 4, 4, 4, 4, 4, 4, 3, 4, 2, 3, 2, 2, 1, 1, 1, 0, 1

Offset: 2

Victor Mishnyakov, Elena Lanina, Apr 25 2020

			Irregular triangle begins:
----------------------------------------------------------
n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
----------------------------------------------------------
  2 | 1 0 1
  3 | 1 0 1 1 0 1
  4 | 1 0 1 1 1 1 1 1 0 1
  5 | 1 0 1 1 1 2 1 2 1 2  1  1  1  0  1
  6 | 1 0 1 1 1 2 2 2 2 3  2  3  2  2  2  2  1  1  1  0  1
  ...
For n = 4: T(4,3) = 1 because we have [3], G.f.=1+x^2+x^3+x^4+x^5+x^6+x^7+x^9;
For n = 5: T(5,5) = 2 because we have [5] and [3,2].
G.f. is 1+x^2+x^3+x^4+2x^5+x^6+2x^7+x^8+2x^9+x^10+x^11+x^12+x^14.

Cf. A000009, A000041, A025147, A087897, A334305.

trow[n_] := CoefficientList[Product[(1 + x^i), {i, 2, n}], x]; nmax = 10; Table[trow[n], {n, 2, nmax}] // Flatten

G.f. for row n: Product_{i=2..n} (1+x^i), n >= 2.

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A362829 Positions in lexicographic order of odd partitions of sufficiently large numbers.

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A115604 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts in which the smallest part occurs k times (1<=k<=n).

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A116663 Triangle read by rows: T(n,k) = number of partitions of n into odd parts and having exactly k parts equal to 1 (n>=0, 1<=k<=n).

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A303902 Expansion of (1 - x^2)*Product_{k>=2} (1 + x^k)^k.

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Mathematica

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A331980 Number of compositions (ordered partitions) of n into distinct odd parts > 1.

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A334377 Irregular triangle read by rows: T(n,k) is the number of partitions of k into distinct parts p such that 2 <= p <= n.

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