A052467 -id:A052467 - cp's OEIS frontend

A103198 Number of compositions of n into a square number of parts.

1, 1, 1, 1, 2, 5, 11, 21, 36, 58, 94, 166, 331, 716, 1574, 3368, 6892, 13447, 25127, 45391, 80428, 142615, 259085, 491855, 982400, 2045001, 4352661, 9291361, 19609786, 40574017, 81973315, 161568281, 311062991, 586764281, 1089615033, 2005257849, 3688711427

Offset: 0

Vladeta Jovovic, Mar 18 2005

From Gus Wiseman, Jan 17 2019: (Start)
Also the number of ways to fill a square matrix with the parts of an integer partition of n. For example, the a(6) = 11 matrices are:
  [6]
.
  [1 1] [1 1] [1 3] [3 1] [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
  [1 3] [3 1] [1 1] [1 1] [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..3329 (terms n = 1..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, a(n+1)/a(n) as a graph

Cf. A000290, A011782, A052467, A089299, A089333, A120732, A323433, A323519, A323525.

b:= proc(n, t) option remember; `if`(n=0,
      `if`(issqr(t), 1, 0), add(b(n-j, t+1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 18 2019

nmax = 40; Rest[CoefficientList[Series[-1/2 + EllipticTheta[3, 0, x/(1-x)]/2, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 03 2017 *)

a(n) = Sum_{k>=0} (x/(1-x))^(k^2).
Binomial transform of the characteristic function of squares A010052, with 0th term omitted. - Carl Najafi, Sep 09 2011
a(n) = Sum_{k >= 0} binomial(n-1,k^2-1). - Gus Wiseman, Jan 17 2019

a(0)=1 prepended by Alois P. Heinz, Jan 18 2019

A121497 Binomial transform of the characteristic function of the prime numbers (A010051).

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 41, 78, 148, 282, 537, 1013, 1882, 3446, 6267, 11468, 21416, 41209, 81771, 166042, 340994, 700570, 1429375, 2886777, 5771828, 11453105, 22638215, 44742141, 88681674, 176545766, 352992931, 707922077, 1421120880, 2849433326

Offset: 0

T. D. Noe, Aug 03 2006

nonn

This is the binomial transform of the sequence {0,0,1,1,0,1,0,1,...}. Sequence A052467, the binomial transform of the sequence {0,1,1,0,1,0,1,...} is very similar. In fact, the first differences of this sequence yields A052467.
The number of pernicious numbers (A052294) less than 2^n. Although the graph looks almost like 2^n, the graph of a(n)/2^n has quite a bit of variation. - T. D. Noe, Mar 14 2009
a(n)/2^n is the probability that a series of Bernoulli trials with probability of success equal to 1/2 will result in a prime number of successes. Cf. A178851. - Eric M. Schmidt, Jul 13 2012
a(n) equals the number of subsets of [n] whose cardinalities are prime. - Ivan N. Ianakiev, Jul 14 2019
Upper and lower bounds are provided by Kim and Sinha (see links). - Jeffrey Shallit, Nov 14 2024

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 0..3324 (terms up to 1000 from Noe)
Sungjin Kim and Nilotpal Kanti Sinha, Binomial probability of prime number of successes, INTEGERS 20 (2020), #A99.
Vaclav Kotesovec, Plot of a(n) / (2^n/log(n/2)) for n = 2..10000

Cf. A000720, A010051, A052294, A052467, A178851

Primes:= select(isprime, [2,seq(i,i=3..100,2)]):
G:= add((z/(1-z))^p/(1-z),p=Primes):
S:= series(G,z,101):
seq(coeff(S,z,i),i=0..100); # Robert Israel, Sep 27 2018

Table[Sum[Binomial[n,Prime[i]], {i,PrimePi[n]}], {n,40}]

a(n)=my(s);forprime(p=2,n,s+=binomial(n,p));s \\ Charles R Greathouse IV, Mar 22 2013

a(n) = Sum_{i=1..pi(n)} binomial(n,prime(i)), where pi(n) is A000720(n), the number of primes <= n.
E.g.f.: exp(x) * (x^2/2! + x^3/3! + x^5/5! + ...) - Eric M. Schmidt, Jul 14 2012
G.f.: Sum_{p prime} x^p/(1-x)^(p+1). - Robert Israel, Sep 27 2018

a(0) inserted by Franklin T. Adams-Watters, Jul 13 2012

A280351 Expansion of Sum_{k>=0} (x/(1 - x))^(k^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1717, 3433, 6436, 11441, 19449, 31825, 50389, 77521, 116281, 170545, 245158, 346105, 480701, 657802, 888058, 1184419, 1564435, 2063206, 2799487, 4272049, 8544097, 23535821, 77331981, 262534537, 865287625, 2720095405

Offset: 0

Ilya Gutkovskiy, Jan 01 2017

Number of compositions of n into a cube number of parts.

			a(9) = 9 because we have:
[1]  [9]
[2]  [2, 1, 1, 1, 1, 1, 1, 1]
[3]  [1, 2, 1, 1, 1, 1, 1, 1]
[4]  [1, 1, 2, 1, 1, 1, 1, 1]
[5]  [1, 1, 1, 2, 1, 1, 1, 1]
[6]  [1, 1, 1, 1, 2, 1, 1, 1]
[7]  [1, 1, 1, 1, 1, 2, 1, 1]
[8]  [1, 1, 1, 1, 1, 1, 2, 1]
[9]  [1, 1, 1, 1, 1, 1, 1, 2]

Index entries for sequences related to compositions

Cf. A000578, A052467, A103198, A280352.

a := n -> ifelse(n = 0, 1, add(binomial(n - 1, k^3 - 1), k = 1..floor(n^(1/3)))):
seq(a(n), n = 0..39); # Peter Luschny, Dec 23 2022

nmax = 39; CoefficientList[Series[Sum[(x/(1 - x))^k^3, {k, 0, nmax}], {x, 0, nmax}], x]

a(0) = 1; a(n) = Sum_{k=1..floor(n^(1/3))} binomial(n-1, k^3-1) for n > 0. - Jerzy R Borysowicz, Dec 22 2022

a(0)=1 prepended by Alois P. Heinz, Dec 17 2022

A280352 Expansion of Sum_{k>=1} (x/(1 - x))^(k*(k+1)/2).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 43, 85, 164, 308, 573, 1079, 2081, 4097, 8129, 16049, 31315, 60402, 115806, 222416, 430791, 843987, 1670054, 3322167, 6606936, 13078586, 25714238, 50230292, 97708338, 189921842, 370216757, 725680489, 1431888173, 2842060970, 5662371069

Offset: 1

Ilya Gutkovskiy, Jan 01 2017

Number of compositions of n into a triangular number of parts.

			a(5) = 7 because we have:
  [1]  [5]
  [2]  [3, 1, 1]
  [3]  [1, 3, 1]
  [4]  [1, 1, 3]
  [5]  [2, 2, 1]
  [6]  [2, 1, 2]
  [7]  [1, 2, 2]

Vaclav Kotesovec, Plot of a(n) / (2^(n-1)/sqrt(n)) for n = 1..10000
Index entries for sequences related to compositions

Cf. A000217, A052467, A103198, A280351.

nmax = 36; Rest[CoefficientList[Series[Sum[(x/(1 - x))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 40; Rest[CoefficientList[Series[-1 + EllipticTheta[2, 0, Sqrt[x/(1-x)]]/(2*(x/(1-x))^(1/8)), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 01 2017 *)

a(n) = sum(k=1, (sqrtint(8*n+1)-1)\2, binomial(n-1, k*(k+1)/2-1)) \\ Andrew Howroyd, Jan 14 2023

G.f.: Sum_{k>=1} (x/(1-x))^(k*(k+1)/2).
a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} binomial(n-1, k*(k+1)/2-1). - Jerzy R Borysowicz, Dec 26 2022
Conjecture: a(n+1)/a(n) ~ 2. - Jerzy R Borysowicz, Jan 14 2023
Conjecture: abs(b(n)-1) < 0.015, where b(n) = a(n)*sqrt(n)/2^(n-1), for n > 781; b(n) does not have a limit. - Jerzy R Borysowicz, Feb 17 2023

A102623 Number of compositions into a prime number of distinct parts.

Original entry on oeis.org

0, 0, 2, 2, 4, 10, 12, 18, 26, 32, 40, 52, 60, 72, 206, 218, 352, 490, 744, 1002, 1382, 1760, 2380, 3004, 3864, 4728, 5954, 12218, 13804, 20554, 27660, 39930, 52682, 75632, 99184, 132940, 172332, 227088, 287606, 373562, 465280, 587602, 725880, 899802, 1094846

Offset: 1

Vladeta Jovovic, Jan 31 2005

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A085756, A052467, A038499.

b:= proc(n, i) option remember; `if`(n=0, [1],
      `if`(n>i*(i+1)/2, [], zip((x, y)->x+y, b(n, i-1),
      `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
a:= proc(n) local l; l:= b(n$2);
      add(`if`(isprime(i), l[i+1]*i!, 0), i=2..nops(l)-1)
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 20 2012

CoefficientList[ Series[ Sum[ Prime[k]!* x^(Prime[k]^2/2 + Prime[k]/2)/Product[1 - x^j, {j, Prime[k]}], {k, 44}], {x, 0, 44}], x] (* Robert G. Wilson v, Feb 04 2005 *)

G.f.: Sum(prime(k)!*x^(1/2*prime(k)^2+1/2*prime(k))/Product(1-x^j, j = 1 .. prime(k)), k = 1 .. infinity).

More terms from Robert G. Wilson v, Feb 04 2005

A339434 Number of compositions (ordered partitions) of n into a prime number of distinct prime parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 2, 2, 2, 8, 0, 8, 2, 8, 8, 10, 0, 16, 8, 16, 14, 16, 12, 18, 14, 22, 18, 136, 18, 138, 26, 22, 26, 258, 30, 266, 30, 266, 158, 492, 36, 506, 158, 510, 278, 744, 174, 748, 290, 758, 528, 990, 306, 1228, 668, 1116, 780, 6384, 678, 6630, 800, 1720, 1274

Offset: 0

Ilya Gutkovskiy, Dec 04 2020

			a(10) = 8 because we have [7, 3], [3, 7], [5, 3, 2], [5, 2, 3], [3, 5, 2], [3, 2, 5], [2, 5, 3] and [2, 3, 5].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to compositions

Cf. A000040, A045450, A052467, A085755, A085756, A102623, A219107.

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0,
         b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 04 2020

s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n - 1]];
b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0,
     If[n == 0, If[PrimeQ[t], t!, 0], Function[p, If[p > n, 0,
       b[n - p, i - 1, t + 1]]][Prime[i]] + b[n, i - 1, t]]];
a[n_] := b[n, PrimePi[n], 0];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

A103197 Number of compositions of n into Fibonacci number of parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 21, 37, 65, 115, 208, 386, 727, 1367, 2536, 4630, 8376, 15217, 28170, 53620, 104843, 208547, 416448, 824990, 1608138, 3071813, 5747106, 10561032, 19177849, 34734782, 63495907, 118601911, 228454377, 454988025, 932297291

Offset: 1

Vladeta Jovovic, Mar 18 2005

Cf. A102848, A052467.

Flatten[{1,2,4,Table[SeriesCoefficient[Sum[(x/(1-x))^Fibonacci[k],{k,2,n}],{x,0,n}],{n,4,35}]}] (* Vaclav Kotesovec, May 01 2014 *)

a(n) = Sum_{k=0..n} binomial(n-1, k-1)*A010056(k).

a(n) = Sum_{k>1} (x/(1-x))^Fibonacci(k).

cp's OEIS Frontend

A103198 Number of compositions of n into a square number of parts.

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A121497 Binomial transform of the characteristic function of the prime numbers (A010051).

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A280351 Expansion of Sum_{k>=0} (x/(1 - x))^(k^3).

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A280352 Expansion of Sum_{k>=1} (x/(1 - x))^(k*(k+1)/2).

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A102623 Number of compositions into a prime number of distinct parts.

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A339434 Number of compositions (ordered partitions) of n into a prime number of distinct prime parts.

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A103197 Number of compositions of n into Fibonacci number of parts.

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