A070933 -id:A070933 - cp's OEIS frontend

A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

1, 1, 2, 4, 12, 30, 66, 164, 419, 1049, 2625, 6372, 15451, 37335, 89855, 216523, 518714, 1235897, 2930050, 6911149, 16217817, 37914515, 88304358, 204971388, 474172899, 1093547574, 2513959446, 5761735383, 13165908506, 29998936859, 68164839887, 154478212575

Offset: 0

Gus Wiseman, Jul 01 2022

nonn

If a composition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of compositions.

			The initial terms count the following containments:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)
                (11)(1)  (21)(21)  (31)(31)
                         (12)(12)  (13)(13)
                         (111)(1)  (22)(2)
                                   (211)(11)
                                   (211)(21)
                                   (121)(11)
                                   (121)(12)
                                   (121)(21)
                                   (112)(11)
                                   (112)(12)
                                   (1111)(1)

Christian Sievers, Table of n, a(n) for n = 0..59

The homog. case is A032020, w/o containment A355388 (partitions A355385).
For partitions we have A355383, homog. A000009, w/ multiplicity A339006.
A000244 counts splittings of compositions, for partitions A323583.
Cf. A001970, A022811, A063834, A070933, A072706, A133494, A336139.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,5}]

a(21) and beyond from Christian Sievers, May 08 2025

A265951 Expansion of Product_{k>=1} 1/(1 - 2kx^k).

Original entry on oeis.org

1, 2, 8, 22, 68, 170, 484, 1166, 3048, 7274, 18000, 41806, 100684, 229258, 535692, 1206230, 2758944, 6123650, 13798088, 30284894, 67272756, 146426002, 321513284, 693944814, 1510245960, 3236648578, 6985572672, 14885926182, 31904642348, 67618415690

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3280

Cf. A006906, A265974, A265975, A265976.

Cf. A032309, A070933, A265955.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      2^n, b(n, i-1) +i*2*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Aug 23 2019

nmax=40; CoefficientList[Series[Product[1/(1-2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * n * 2^n, where c = 1/2 * Product_{m>=3} 1/(1 - m/2^(m-1)) = 9.281573281805057363737677116134642024212942973614535341005126953773818...

A303346 Expansion of Product_{n>=1} ((1 + 2x^n)/(1 - 2x^n))^(1/2).

Original entry on oeis.org

1, 2, 4, 10, 18, 38, 72, 142, 260, 510, 940, 1814, 3362, 6490, 12112, 23466, 44114, 85766, 162516, 317190, 604806, 1184682, 2271248, 4461514, 8591784, 16916490, 32696708, 64496130, 125037142, 247007142, 480077432, 949510526, 1849375796, 3661330398, 7144215452

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A032302, A070933, A261584, A303307, A303345, A370709, A370713.

nmax = 30; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[Sqrt[-QPochhammer[-2, x] / (3*QPochhammer[2, x])], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+2*x^k)/(1-2*x^k))^(1/2)))

a(n) ~ 2^n / sqrt(c*Pi*n), where c = A048651 * A083864 = 1/2 * Product_{j>=1} (2^j-1)/(2^j+1) = 0.06056210400129025123042464659093375290492912341... - Vaclav Kotesovec, Apr 22 2018

A070877 Expansion of Product_{k>=1} (1 - 2x^k).

Original entry on oeis.org

1, -2, -2, 2, 2, 6, -2, 2, -6, -10, -2, -6, -6, 6, 22, -6, 26, 14, 22, -6, -14, -2, -10, -46, -46, -50, -18, 18, -78, 22, 14, 82, 42, 166, 14, 42, 170, 118, 6, 106, -150, -66, -122, -118, -62, -370, -282, -350, -126, -354, -2, -94, 226, -250, 30, 450, 730, 342, 894, 474, 890, 358, 758, 58, 1210, 782, -778, 26, -270, -1250

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002

sign

			G.f. = 1 - 2*x - 2*x^2 + 2*x^3 + 2*x^4 + 6*x^5 - 2*x^6 + 2*x^7 - 6*x^8 - 10*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Giovanni Resta)

Cf. A070933, A010815, A032302.

CoefficientList[ Series[ Product[(1 - 2t^k), {k, 1, 80}], {t, 0, 80}], t]
a[ n_] := SeriesCoefficient[ -QPochhammer[2, x], {x, 0, n}]; (* Michael Somos, Mar 11 2014 *)

N=66; q='q+O('q^N); Vec(sum(n=0, N, (-2)^n*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ Joerg Arndt, Mar 09 2014

N=66; q='q+O('q^N); t2=Vec( prod(k=1, N, 1-2*q^k) ) \\ Joerg Arndt, Mar 11 2014

Edited by Robert G. Wilson v, May 26 2002

Corrected by Vincenzo Librandi, Mar 11 2014

A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 16, 26, 42, 64, 100, 150, 224, 330, 482, 697, 999, 1418, 1996, 2794, 3879, 5355, 7343, 10018, 13583, 18338, 24618, 32917, 43790, 58043, 76591, 100716, 131906, 172194, 223966, 290423, 375318, 483668, 621368, 796138, 1017146

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

If a partition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of partitions.

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(1)  (21)(21)  (31)(31)   (41)(41)
                         (111)(1)  (22)(2)    (32)(32)
                                   (211)(11)  (311)(11)
                                   (211)(21)  (311)(31)
                                   (1111)(1)  (221)(21)
                                              (221)(22)
                                              (2111)(11)
                                              (2111)(21)
                                              (11111)(1)

With multiplicity we have A339006.
The version for compositions is A355384.
The homogeneous version w/o containment is A355385, compositions A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Splitting partitions: A072706, A316245, A317715, A318683, A318684, A319794, A323433, A323583, A336131-A336136.
Cf. A000009, A022811, A032020, A070933, A181591, A181819, A319910, A355382.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,IntegerPartitions[n]}],{n,0,15}]

A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 25, 43, 81, 141, 243, 409, 699, 1132, 1844, 2995, 4744, 7408, 11655, 17839, 27509, 41546, 62879, 93537, 139974, 205547, 302714, 440097, 640968, 921774, 1327538, 1891548, 2696635, 3809860, 5380257, 7540778, 10561566, 14687109, 20408170, 28183998, 38882009

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

Also the number of composable pairs of integer partitions of n, where a partition is regarded as an arrow from (number of parts) to (number of distinct parts). Is there a nice choice of a composition operation making this into an associative category?

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(2)  (21)(21)  (31)(31)   (41)(41)
                         (111)(3)  (31)(22)   (41)(32)
                                   (22)(4)    (32)(41)
                                   (211)(31)  (32)(32)
                                   (211)(22)  (311)(41)
                                   (1111)(4)  (311)(32)
                                              (221)(41)
                                              (221)(32)
                                              (2111)(41)
                                              (2111)(32)
                                              (11111)(5)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The inhomogeneous version with containment and multiplicity is A339006.
The inhomogeneous version with containment is A355383.
The inhomogeneous version with containment for compositions is A355384.
The version for compositions is A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
A323583 counts splittings of partitions.
Cf. A000009, A008284, A022811, A032020, A070933, A072706, A116608, A279787, A319910, A336135.

Table[Length[Select[Tuples[IntegerPartitions[n],2],Length[Union[#[[1]]]]==Length[#[[2]]]&]],{n,0,15}]

\\ P gives A008284 and R gives A116608 as g.f.'s.
P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(n,y) = {prod(k=1, n, 1 + y/(1 - x^k) - y + O(x*x^n))}
seq(n) = {my(g=Vec(P(n,y)), h=Vec(R(n,y))); vector(n+1, i, my(p=g[i], q=h[i]); sum(j=0, poldegree(q), polcoef(p,j)*polcoef(q,j)))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{j >= 1} A116608(n,j) * A008284(n,j) for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A355388 Number of composable pairs (y, v) of integer compositions of n, where a composition is regarded as an arrow from the number of parts to the number of distinct parts.

Original entry on oeis.org

1, 1, 2, 6, 18, 58, 174, 536, 1656, 4947, 14800, 43157, 126572, 364070, 1039926, 2938898, 8223400, 22846370, 62930113, 172177400, 467002792, 1259736804, 3371190792, 8973530491, 23728305128, 62421018163, 163255839779, 424842462529, 1100006243934, 2834558927244, 7270915592897

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

Being composable here means that the length of v equals the number of distinct parts in y.

			The a(0) = 1 through a(4) = 18 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)
                (11)(2)  (21)(21)  (31)(31)
                         (21)(12)  (31)(13)
                         (12)(21)  (31)(22)
                         (12)(12)  (13)(31)
                         (111)(3)  (13)(13)
                                   (13)(22)
                                   (22)(4)
                                   (211)(31)
                                   (211)(13)
                                   (211)(22)
                                   (121)(31)
                                   (121)(13)
                                   (121)(22)
                                   (112)(31)
                                   (112)(13)
                                   (112)(22)
                                   (1111)(4)

Alois P. Heinz, Table of n, a(n) for n = 0..800 (first 201 terms from Andrew Howroyd)

The case with containment is A032020.
The inhomogeneous version with containment is A355384, partitions A355383.
The version for partitions is A355385, with containment A000009.
A133494 counts compositions of each part of a composition, strict A336139.
A323583 counts splittings of partitions.
Cf. A001970, A022811, A063834, A070933, A316245, A319910, A355382.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      expand(add(b(n-i*j, i-1, p+j)/j!*`if`(j=0, 1, x), j=0..n/i))))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(n-1, i-1), i=0..degree(p)))(b(n$2, 0)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 01 2023

Table[Length[Select[Tuples[Join@@Permutations/@IntegerPartitions[n],2], Length[Union[#[[1]]]]==Length[#[[2]]]&]],{n,0,10}]

a(n) = {if(n==0, 1, my(s=0); forpart(p=n, p=Vec(p); my(S=Set(p)); s += binomial(n-1, #S-1)*(#p)!/prod(i=1, #S, my(c=#select(t->t==S[i], p)); c! )); s)} \\ Andrew Howroyd, Jan 01 2023

\\ for larger n.
a(n) = { local(Cache=Map());
  my(F(r,m,p,q) = my(key=[r,m,p,q], z); if(!mapisdefined(Cache, key, &z),
  z = if(m==0, if(r==0, p!*binomial(n-1, q-1)), self()(r, m-1, p, q) + sum(j=1, r\m, self()(r-j*m, min(m-1, r-j*m), p+j, q+1)/j!));
  mapput(Cache, key, z) ); z);
  if(n==0, 1, F(n, n, 0, 0))
} \\ Andrew Howroyd, Jan 01 2023

a(n) = Sum_{k>=1} binomial(n-1, k-1)*A235998(n, k) for n > 0. - Andrew Howroyd, Jan 01 2023

Terms a(14) and beyond from Andrew Howroyd, Jan 01 2023

A071109 Expansion of Product_{k>=1} 1/(1+2*x^k).

Original entry on oeis.org

1, -2, 2, -6, 14, -26, 50, -102, 214, -426, 834, -1678, 3398, -6778, 13482, -27022, 54198, -108306, 216346, -432878, 866334, -1732386, 3463626, -6927926, 13858350, -27715378, 55426002, -110855030, 221719582, -443433610, 886848930, -1773709078, 3547455846

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 27 2002

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A070933, A032302, A000700, A246935, A261562, A261566, A261582.

nmax = 40; CoefficientList[Series[Product[1/(1 + 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^k*2^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(O[x]^30 + 3/QPochhammer[-2, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) ~ c * (-2)^n, where c = Product_{j>=1} 1/(1-1/(-2)^j) = 1/QPochhammer[-1/2,-1/2] = 0.8259519860658427384636116224100201356301... . - Vaclav Kotesovec, Aug 25 2015

G.f.: Sum_{i>=0} (-2)^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 13 2018

More terms from Vaclav Kotesovec, Aug 25 2015

A261561 Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.

Original entry on oeis.org

1, 2, 8, 22, 64, 162, 424, 1022, 2480, 5770, 13336, 30046, 67184, 147554, 321592, 692278, 1479568, 3133474, 6596008, 13788606, 28679264, 59335530, 122256456, 250875550, 513116864, 1046190786, 2127557592, 4316282006, 8739096992, 17661731138, 35639764536

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Vaclav Kotesovec)

Cf. A000219, A070933, A246935, A261562, A261563, A261565.

b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
      add(2^j*binomial(i+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 21 2018

nmax = 50; CoefficientList[Series[Product[(1/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

{a(n) = polcoeff( exp( sum(m=1,n,x^m/m * sumdiv(m,d,2^d*m^2/d^2) ) +x*O(x^n)),n)}
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Sep 30 2015

a(n) ~ c * 2^n, where c = Product_{j>=1} 1/(1 - 1/2^j)^(j+1) = 34.7387234654851595844514193757064296508992247003230539635669599773458896...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} 2^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

A370713 a(n) = 2^n * [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/2).

Original entry on oeis.org

1, 2, 10, 36, 150, 524, 2116, 7528, 29158, 107404, 409388, 1526712, 5843580, 22048056, 84476488, 322465104, 1240657798, 4769404556, 18436257820, 71280387736, 276583603220, 1074167182184, 4181581140280, 16295088132272, 63609310195164, 248542739388472, 972391744910392

Offset: 0

Vaclav Kotesovec, Feb 27 2024

nonn

Cf. A070933, A370709.

nmax = 35; CoefficientList[Series[Product[1/(1-2*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]
nmax = 35; CoefficientList[Series[Product[1/(1-2*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Sqrt[-1/QPochhammer[2,x]], {x, 0, nmax}], x] * 2^Range[0, nmax]

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

a(n) ~ 4^n / sqrt(Pi*QPochhammer(1/2)*n).

cp's OEIS Frontend

A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

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A265951 Expansion of Product_{k>=1} 1/(1 - 2*k*x^k).

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A303346 Expansion of Product_{n>=1} ((1 + 2*x^n)/(1 - 2*x^n))^(1/2).

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A070877 Expansion of Product_{k>=1} (1 - 2x^k).

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A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

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A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.

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A355388 Number of composable pairs (y, v) of integer compositions of n, where a composition is regarded as an arrow from the number of parts to the number of distinct parts.

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A265951 Expansion of Product_{k>=1} 1/(1 - 2kx^k).

A303346 Expansion of Product_{n>=1} ((1 + 2x^n)/(1 - 2x^n))^(1/2).