A075900 -id:A075900 - cp's OEIS frontend

A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

1, 2, 2, 20, 6, 108, 148, 776, -186, 5964, -4, 51032, -89700, 512120, -1259416, 6406032, -19733434, 78363148, -268823572, 1047941688, -3800035916, 14327505832, -52766730600, 199492430192, -746479735524, 2811936761016, -10588174502568, 40092283176560, -151796846803592

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032302, A370713.

Cf. A075900, A298994, A300581, A304961, A327550.

nmax = 30; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[(1 + 2*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Sqrt[QPochhammer[-2, x]/3], {x, 0, nmax}], x] * 2^Range[0, nmax]

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

a(n) ~ (-1)^(n+1) * c * 4^n / n^(3/2), where c = QPochhammer(-1/2)^(1/2) / (2*sqrt(Pi)) = 0.31039710860287467176143051675437...

A370716 a(n) = 3^(2n) [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).

Original entry on oeis.org

1, 6, 18, 1170, -1890, 133326, 101250, 20498994, -164656314, 3778220862, -28085954094, 771567716970, -10691904063114, 183594050113518, -2711145260068326, 49416883617381354, -789899109743435994, 13176840267952166070, -216403389726994588086, 3681309971143060236810

Offset: 0

Vaclav Kotesovec, Feb 27 2024

sign

Cf. A032302, A370715.

Cf. A075900, A300581, A304961, A327550.

nmax = 20; CoefficientList[Series[Product[(1 + 2*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 3^(2*Range[0, nmax])
nmax = 20; CoefficientList[Series[Product[(1 + 2*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[(QPochhammer[-2, x]/3)^(1/3), {x, 0, nmax}], x] * 3^(2*Range[0, nmax])

G.f.: Product_{k>=1} (1 + 2*(9*x)^k)^(1/3).

a(n) ~ (-1)^(n+1) * c * 18^n / n^(4/3), where c = QPochhammer(-1/2)^(1/3) / (3*Gamma(2/3)) = 0.2623638446186535909018671540030519...

A080267 a(n) = Sum_{d divides n} d*2^(n-n/d).

Original entry on oeis.org

1, 5, 13, 41, 81, 257, 449, 1313, 2497, 6465, 11265, 33665, 53249, 143617, 269313, 672257, 1114113, 3159041, 4980737, 13568001, 23904257, 57675777, 96468993, 275980289, 424673281, 1090535425, 1963720705, 4823482369, 7784628225

Offset: 1

Vladeta Jovovic, Feb 11 2003

nonn

Robert Israel, Table of n, a(n) for n = 1..3307

Cf. A075900, A359112.

oo := 40; s1 := add( k*2^(k-1)*x^k/(1-2^(k-1)*x^k),k=1..oo): s2 := series(s1,x,oo-1): s3 := seriestolist(%): A080267 := n->s3[n+1];

a[n_] := Sum[d*2^(n-n/d), {d, Divisors[n]}]; Array[a, 29] (* Jean-François Alcover, Mar 20 2014 *)

a(n) = sumdiv(n, d, d*2^(n-n/d)); \\ Michel Marcus, Mar 20 2014

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-(2*x)^k)^2)) \\ Seiichi Manyama, Dec 20 2022

G.f.: Sum_{k>=1} k*2^(k-1)*x^k/(1 - 2^(k-1)*x^k). - N. J. A. Sloane, Jun 04 2003

G.f.: Sum_{k>=1} x^k/(1 - (2 * x)^k)^2. - Seiichi Manyama, Dec 20 2022

A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 17, 22, 26, 35, 40, 51, 60, 75, 86, 109, 124, 153, 175, 214, 243, 297, 336, 403, 456, 546, 614, 731, 821, 975, 1095, 1283, 1437, 1689, 1887, 2195, 2448, 2851, 3172, 3676, 4083, 4724, 5245, 6022, 6677, 7695, 8504, 9720

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(9) = 9 splittings:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)    (5,4)
                          (4,1)  (5,1)    (5,2)    (6,2)    (6,3)
                                 (3,2,1)  (6,1)    (7,1)    (7,2)
                                          (4,2,1)  (4,3,1)  (8,1)
                                                   (5,2,1)  (4,3,2)
                                                            (5,3,1)
                                                            (6,2,1)
                                                            (4),(3,2)
The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal sums is A318683.
The version with strictly decreasing sums is A318684.
The version with weakly decreasing sums is A319794.
The version with different sums is A336132.
Starting with a composition gives A304961.
Starting with a non-strict partition gives A336134.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336127, A336128, A336130, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 31, 40, 73, 98, 158, 204, 340, 420, 629, 819, 1202, 1494, 2174, 2665, 3759, 4688, 6349, 7806, 10788, 13035, 17244, 21128, 27750, 33499, 43941, 52627, 67957, 81773, 103658, 124047, 158628, 187788, 235162, 280188, 349612, 413120, 513952, 604568

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(5) = 15 splittings:
  (1)  (2)      (3)          (4)              (5)
       (1,1)    (2,1)        (2,2)            (3,2)
       (1),(1)  (1,1,1)      (3,1)            (4,1)
                (1),(1,1)    (2,1,1)          (2,2,1)
                (1),(1),(1)  (2),(2)          (3,1,1)
                             (1,1,1,1)        (2,1,1,1)
                             (2),(1,1)        (2),(2,1)
                             (1),(1,1,1)      (1,1,1,1,1)
                             (1,1),(1,1)      (2),(1,1,1)
                             (1),(1),(1,1)    (1),(1,1,1,1)
                             (1),(1),(1),(1)  (1,1),(1,1,1)
                                              (1),(1),(1,1,1)
                                              (1),(1,1),(1,1)
                                              (1),(1),(1),(1,1)
                                              (1),(1),(1),(1),(1)

Andrew Howroyd, Table of n, a(n) for n = 0..60

The version with weakly decreasing sums is A316245.
The version with equal sums is A317715.
The version with strictly increasing sums is A336134.
The version with strictly decreasing sums is A336135.
The version with different sums is A336131.
Starting with a composition gives A075900.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A304961, A305551, A318684, A323433, A336128, A336130, A336133.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],LessEqual@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]

a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r >= t && t >= s, self()(r,m,t,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m),s,t+m,1))); recurse(n,n,0,0)} \\ Andrew Howroyd, Jan 18 2024

a(21) onwards from Andrew Howroyd, Jan 18 2024

A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 22, 41, 72, 142, 260, 454, 769, 1416, 2472, 4465, 7708, 13314, 23630, 40406, 68196, 119646, 203237, 343242, 586508, 993764, 1677187, 2824072, 4753066, 7934268, 13355658, 22229194, 36945828, 61555136, 102019156, 168474033, 279181966

Offset: 0

Gus Wiseman, Jul 18 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (1,2)    (1,3)      (1,4)
            (2,1)    (3,1)      (2,3)
            (2),(1)  (3),(1)    (3,2)
                     (1,2),(1)  (4,1)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1,2),(2)
                                (1,3),(1)
                                (2,1),(2)
                                (3,1),(1)

Alois P. Heinz, Table of n, a(n) for n = 0..7725

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 31 2020

strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[Join@@Permutations/@strptn[#]&/@ctn],{ctn,strptn[n]}]],{n,0,20}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
     If[n == 0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0,
     If[n == 0, 1, g[n, i-1] + b[i, i, 0]*g[n-i, Min[n-i, i-1]]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

G.f.: Product_{k >= 1} (1 + A032020(k)*x^k).

A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^k).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 31, 57, 113, 212, 410, 757, 1464, 2684, 5083, 9380, 17569, 32120, 59977, 109193, 202046, 367951, 675541, 1224453, 2243795, 4052369, 7377243, 13314989, 24140198, 43406515, 78510429, 140800279, 253663615, 454352111, 815790813, 1457485309

Offset: 0

Vaclav Kotesovec, Mar 08 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500

Cf. A000045, A023882, A075900, A077365, A179381, A318248.

nmax = 40; CoefficientList[Series[Product[1/(1-Fibonacci[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

log(a(n)) ~ log(phi)*n + 2*sqrt(polylog(2, 1/sqrt(5))*n) - 3*(log(n)/4), where polylog(2, 1/sqrt(5)) = 0.5107013915606224266804289751265205446721... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio.

A327548 Total number of compositions in the compositions of partitions of n.

Original entry on oeis.org

0, 1, 4, 11, 34, 85, 248, 603, 1630, 4017, 10308, 24855, 63210, 150141, 369936, 882083, 2135606, 5023689, 12064092, 28167919, 66828418, 155569685, 364983208, 844175675, 1971322574, 4533662817, 10498550260, 24077361031, 55432615194, 126492183213, 289997946944

Offset: 0

Alois P. Heinz, Sep 16 2019

nonn

			a(3) = 11 = 1+1+1+1+2+2+3 counts the compositions in 3, 21, 12, 111, 2|1, 11|1, 1|1|1.

Alois P. Heinz, Table of n, a(n) for n = 0..3202

Cf. A000041, A011782, A075900, A326346, A327549.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      b(n, i-1)+(p->p+[0, p[1]])(2^(i-1)*b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..32);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, b[n, i - 1] + With[{p = 2^(i - 1) b[n - i, Min[n - i, i]]}, p + {0, p[[1]]}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A327549(n,k).

a(n) ~ log(2) * (3/(Pi^2 - 6*log(2)^2))^(1/4) * 2^(n-1) * exp(sqrt((Pi^2 - 6*log(2)^2)*n/3)) / (sqrt(Pi) * n^(1/4)). - Vaclav Kotesovec, Sep 19 2019

A327549 Number T(n,k) of compositions of partitions of n with exactly k compositions; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 8, 8, 2, 1, 0, 16, 16, 8, 2, 1, 0, 32, 48, 24, 8, 2, 1, 0, 64, 96, 64, 24, 8, 2, 1, 0, 128, 256, 160, 80, 24, 8, 2, 1, 0, 256, 512, 448, 192, 80, 24, 8, 2, 1, 0, 512, 1280, 1024, 576, 224, 80, 24, 8, 2, 1

Offset: 0

Alois P. Heinz, Sep 16 2019

			T(3,1) = 4: 3, 21, 12, 111.
T(3,2) = 2: 2|1, 11|1.
T(3,3) = 1: 1|1|1.
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,    1;
  0,   4,    2,    1;
  0,   8,    8,    2,   1;
  0,  16,   16,    8,   2,   1;
  0,  32,   48,   24,   8,   2,  1;
  0,  64,   96,   64,  24,   8,  2,  1;
  0, 128,  256,  160,  80,  24,  8,  2, 1;
  0, 256,  512,  448, 192,  80, 24,  8, 2, 1;
  0, 512, 1280, 1024, 576, 224, 80, 24, 8, 2, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-2 give: A000007, A011782 (for n>0), A134353(n-2) (for n>1).
Row sums give A075900.
T(2n,n) gives A327550.
Cf. A060642, A327548.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand(2^(i-1)*x*b(n-i, min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + 2^(i-1) x b[n-i, Min[n-i, i]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 12] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A327548(n).

A337299 Expansion of Product_{k>0} (1 - 2^(k-1)*x^k).

Original entry on oeis.org

1, -1, -2, -2, -4, 0, -8, 16, 0, 64, 64, 384, 0, 1536, 1024, 3072, 2048, 16384, -8192, 49152, -32768, 32768, -65536, 262144, -1835008, 524288, -3145728, -6291456, -18874368, -4194304, -117440512, -16777216, -301989888, -469762048, -671088640, -805306368, -6710886400, 536870912

Offset: 0

Seiichi Manyama, Aug 22 2020

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = 2^(n-1).

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Convolution inverse of A075900.

Cf. A266964, A304961.

m = 37; CoefficientList[Series[Product[1 - 2^(k - 1)*x^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 22 2020 *)

N=40; x='x+O('x^N); Vec(prod(k=1, N, 1-2^(k-1)*x^k))

cp's OEIS Frontend

A370709 a(n) = 2^n * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/2).

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A370716 a(n) = 3^(2*n) * [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).

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A080267 a(n) = Sum_{d divides n} d*2^(n-n/d).

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A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

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A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

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A336142 Number of ways to choose a strict composition of each part of a strict integer partition of n.

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A300520 Expansion of Product_{k>=1} 1 / (1 - Fibonacci(k)*x^k).

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A327548 Total number of compositions in the compositions of partitions of n.

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A370716 a(n) = 3^(2n) [x^n] Product_{k>=1} (1 + 2*x^k)^(1/3).