A022567 -id:A022567 - cp's OEIS frontend

A258797 a(n) = [x^n] Product_{k=1..n} (1+x^k)^2 / x^k.

1, 1, 2, 6, 16, 51, 166, 554, 1896, 6595, 23212, 82582, 296393, 1071738, 3900696, 14278074, 52526972, 194108087, 720197524, 2681854490, 10019539112, 37545876368, 141080872362, 531457445806, 2006678785762, 7593123695669, 28789152013570, 109356019134584

Offset: 0

Vaclav Kotesovec, Jun 10 2015

nonn

a(n) is half the number of subsets of {-n..n} whose sum is n. - Ilya Gutkovskiy, Jul 09 2025

Alois P. Heinz, Table of n, a(n) for n = 0..555

Cf. A022567, A047653, A258798, A258799.

b:= proc(n, s) option remember; `if`(n*(n+1)/2 b(n$2):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 14 2025

Table[SeriesCoefficient[Product[(1+x^k)^2/x^k, {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[Product[1+x^k, {k, 1, n}]^2, {x, 0, n*(n+3)/2}], {n, 0, 30}]

a(n) ~ sqrt(3) * 4^n / (sqrt(Pi) * n^(3/2)).

A280245 Expansion of Product_{k>=1} (1 + x^prime(k))^2.

Original entry on oeis.org

1, 0, 2, 2, 1, 6, 1, 8, 6, 6, 14, 6, 18, 14, 18, 24, 23, 30, 35, 38, 46, 54, 55, 74, 72, 90, 100, 106, 128, 136, 152, 178, 185, 216, 238, 252, 302, 308, 359, 390, 420, 478, 512, 564, 628, 668, 745, 810, 871, 974, 1035, 1140, 1238, 1336, 1459, 1586, 1700, 1868, 1993, 2168, 2354, 2512, 2751, 2930, 3177, 3418, 3677, 3960

Offset: 0

Ilya Gutkovskiy, Dec 29 2016

nonn

Number of partitions of n into distinct prime parts, with 2 types of each part.

Self-convolution of A000586. - Ilya Gutkovskiy, Jan 19 2018

			a(5) = 6 because we have [5], [5'], [3, 2], [3', 2], [3, 2'], [3', 2'].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Prime Partition
Index entries for related partition-counting sequences

Cf. A000009, A000586, A000607, A022567, A070215.

nmax = 67; CoefficientList[Series[Product[(1 + x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x]

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

log(a(n)) ~ 2*Pi*sqrt(n/(3*log(n/2))). - Vaclav Kotesovec, Jan 12 2021

A293377 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji))^2.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 7, 6, 0, 1, 2, 7, 10, 9, 0, 1, 2, 7, 16, 25, 14, 0, 1, 2, 7, 16, 31, 38, 22, 0, 1, 2, 7, 16, 39, 62, 78, 32, 0, 1, 2, 7, 16, 39, 70, 117, 116, 46, 0, 1, 2, 7, 16, 39, 80, 149, 206, 206, 66, 0, 1, 2, 7, 16, 39, 80, 159, 262, 362

Offset: 0

Seiichi Manyama, Oct 07 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0,  2,  2,  2,  2, ...
   0,  3,  7,  7,  7, ...
   0,  6, 10, 16, 16, ...
   0,  9, 25, 31, 39, ...
   0, 14, 38, 62, 70, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A000007, A022567.
Rows n=0 gives A000012.
Main diagonal gives A293378.
Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: A290217 (m=-1), A290216 (m=1), this sequence (m=2).

A304443 Coefficient of x^n in Product_{k>=1} (1+x^k)^(2*n).

Original entry on oeis.org

1, 2, 10, 62, 394, 2562, 16966, 113794, 770442, 5254334, 36042250, 248403586, 1718732998, 11931569028, 83064794746, 579696375972, 4054279504266, 28408328186508, 199390547044342, 1401564307833908, 9865190079554954, 69522550703432476, 490484539061916794

Offset: 0

Vaclav Kotesovec, May 12 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A022567, A026011, A270913, A304444.

nmax = 25; Table[SeriesCoefficient[Product[(1+x^k)^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 25; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) {1/r, Sqrt[Derivative[0, 1][QPochhammer][-1, r*s] / (Pi*r*(Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s]^2 + 2*s*Derivative[0, 2][QPochhammer][-1, r*s]))]} /. FindRoot[{4*s == QPochhammer[-1, r*s]^2, 2*r*Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ c * d^n / sqrt(n), where d = 7.21883059750200610514730564495768943165197819880185778427663522275469... and c = 0.300860732623379969554285615234449502629772950943717460278989499...

A327214 Self-convolution of A270913.

Original entry on oeis.org

1, 2, 7, 32, 137, 592, 2597, 11442, 50567, 224112, 995392, 4428372, 19727877, 87983202, 392755207, 1754625632, 7844003907, 35086658052, 157023432677, 703037135122, 3148915010832, 14108913792342, 63235380631747, 283495965998772, 1271282293531077, 5702105357347602

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A022567, A270913, A304443, A309986, A327215.

A270913[n_]:=SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A270913[k]*A270913[n-k], {k, 0, n}], {n, 0, 25}]

a(n) ~ c^2 * Pi * d^n, where d = A270914 = 4.5024767476173544877385939327... and c = A327280 = 0.260542233142438469433860832160... (see A270913).

A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 0, 3, 19, 101, 501, 2486, 12398, 62329, 315436, 1605330, 8207552, 42124368, 216903051, 1119974861, 5796944342, 30068145889, 156250892593, 813310723907, 4239676354631, 22130265931880, 115654632452514, 605081974091853, 3168828466966365, 16610409114771876, 87141919856550506

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Number of partitions of n into 2 or more parts of n kinds. - Ilya Gutkovskiy, May 16 2018

Index entries for sequences related to partitions

Cf. A000065, A008485, A022567, A093160, A270913, A285927, A285928, A286653, A296044, A296162, A296163, A304626.

Table[SeriesCoefficient[Product[((1 - x^(n k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 25}]

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724... and c = 0.268015212710733315686... - Vaclav Kotesovec, May 16 2018

A307756 Exponential convolution of number of partitions into distinct parts (A000009) with themselves.

Original entry on oeis.org

1, 2, 4, 10, 26, 66, 184, 472, 1268, 3340, 8748, 22772, 59102, 151590, 386830, 983914, 2489384, 6263284, 15703204, 39221884, 97498736, 241538472, 596115898, 1465958522, 3595196600, 8788765304, 21421616934, 52080152238, 126268822824, 305365334180, 736770528064

Offset: 0

Ilya Gutkovskiy, Apr 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A000009, A022567, A266232, A307755.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(binomial(n, j)*b(j)*b(n-j), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 26 2019

nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] PartitionsQ[k] PartitionsQ[n - k], {k, 0, n}], {n, 0, 30}]

E.g.f.: (Sum_{k>=0} A000009(k)*x^k/k!)^2.
a(n) = Sum_{k=0..n} binomial(n,k)*A000009(k)*A000009(n-k).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 2^(n - 5/2) / (sqrt(3) * n^(3/2)). - Vaclav Kotesovec, May 06 2019

A316142 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^k)^2.

Original entry on oeis.org

1, 2, 8, 56, 476, 4832, 58508, 815936, 12750956, 220610432, 4195325708, 86976996416, 1949966347436, 46965887762432, 1208922621624908, 33111231803362496, 961354836530983916, 29490401681798152832, 952900154176192244108, 32342850619899263226176

Offset: 0

Vaclav Kotesovec, Jun 25 2018

Self-convolution of A305550.

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). - Peter Bala, Mar 03 2023

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A022567, A305550, A316143, A316144.

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

Sum_{k=0..n} binomial(n,k) * A305550(k) * A305550(n-k).

a(n) ~ n! * exp(Pi * sqrt(n/(3*log(2))) - Pi^2 * (1 - 1/log(2)) / 24) / (2^(5/2) * 3^(1/4) * (log(2))^(n + 1/4) * n^(3/4)).

A229707 Triangular array read by rows. T(n,k) is the number of strictly unimodal compositions of n with the greatest part equal to k; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 3, 2, 1, 0, 0, 4, 3, 2, 1, 0, 0, 3, 6, 3, 2, 1, 0, 0, 2, 7, 6, 3, 2, 1, 0, 0, 1, 8, 9, 6, 3, 2, 1, 0, 0, 0, 10, 12, 9, 6, 3, 2, 1, 0, 0, 0, 8, 16, 14, 9, 6, 3, 2, 1, 0, 0, 0, 7, 20, 20, 14, 9, 6, 3, 2, 1

Offset: 1

Geoffrey Critzer, Sep 27 2013

A strictly unimodal composition is a composition such that for some j,m  1 <= x(1) < x(2) < ... < x(j) > x(j+1) > ... > x(m) >= 1.
Row sums are A059618.
Sum of column k is A000302(k-1).
T(2*n+1,n+1) = A022567(n) for n>=0. - Alois P. Heinz, Oct 11 2013

			1,
0, 1,
0, 2, 1,
0, 1, 2, 1,
0, 0, 3, 2, 1,
0, 0, 4, 3, 2, 1,
0, 0, 3, 6, 3, 2, 1,
0, 0, 2, 7, 6, 3, 2, 1,
0, 0, 1, 8, 9, 6, 3, 2, 1,
0, 0, 0, 10, 12, 9, 6, 3, 2, 1
T(7,3) = 3 because we have: 1+2+3+1 = 1+3+2+1 = 2+3+2.

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A229706.

b:= proc(n, t, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(k>0, `if`(n b(n, 0, k):
seq(seq(T(n, k), k=1..n), n=1..16);  # Alois P. Heinz, Oct 07 2013

nn=10;Table[Take[Drop[Transpose[Map[PadRight[#,nn+1,0]&,Table[CoefficientList[Series[x^n Product[(1+x^i),{i,1,n-1}]^2,{x,0,nn}],x],{n,1,nn}]]],1][[n]],n],{n,1,nn}]//Grid

O.g.f. for column k: x^k * prod(i=1..k-1, 1 + x^i)^2.

A292445 Expansion of a q-series used by Ramanujan in his Lost Notebook.

Original entry on oeis.org

1, 4, 10, 22, 44, 82, 145, 248, 410, 658, 1036, 1598, 2420, 3614, 5322, 7738, 11132, 15850, 22353, 31260, 43366, 59708, 81650, 110932, 149788, 201112, 268562, 356790, 471732, 620834, 813480, 1061496, 1379626, 1786282, 2304440, 2962566, 3795921, 4848160

Offset: 0

Michael Somos, Sep 16 2017

nonn

Similar to A292420 but with a=1.

			G.f. = 1 + 4*x + 10*x^2 + 22*x^3 + 44*x^4 + 82*x^5 + 145*x^6 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=1.

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A022567, A053253, A256209, A279715, A292420.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 / QPochhammer[ x]^2 Sum[ x^k / Product[ 1 - x^(2 i + 1), {i, 0, k}], {k, 0, n}], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^2 * sum(k=0, n, x^k / prod(i=0, k, 1 - x^(2*i+1), 1 + A/x^k)), n))};

a(n) = 2 * A256209(n) - A279715(n).

G.f. is the product of the g.f. of A022567 and A053253.

cp's OEIS Frontend

A258797 a(n) = [x^n] Product_{k=1..n} (1+x^k)^2 / x^k.

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A280245 Expansion of Product_{k>=1} (1 + x^prime(k))^2.

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A293377 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^2.

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A304443 Coefficient of x^n in Product_{k>=1} (1+x^k)^(2*n).

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A327214 Self-convolution of A270913.

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A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

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A307756 Exponential convolution of number of partitions into distinct parts (A000009) with themselves.

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Maple

Mathematica

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A316142 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^k)^2.

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A229707 Triangular array read by rows. T(n,k) is the number of strictly unimodal compositions of n with the greatest part equal to k; n>=1, 1<=k<=n.

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A293377 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji))^2.