cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A134774 G.f.: A(x) = Product_{n>=1} G(x^n,n)^n where G(x,n) = 1 + x*G(x,n)^n.

Original entry on oeis.org

1, 1, 3, 6, 15, 26, 66, 110, 253, 460, 966, 1680, 3732, 6304, 13073, 23539, 47548, 82362, 171463, 293578, 597934, 1056830, 2105424, 3654919, 7533609, 12915780, 26112978, 46033557, 92504870, 160298673, 330468463, 568239653, 1161488784
Offset: 0

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Author

Paul D. Hanna, Nov 11 2007

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 26*x^5 + 66*x^6 +...
G.f.: A(x) = 1/(1-x) * G(x^2,2)^2 * G(x^3,3)^3 * G(x^4,4)^4 *...
where the functions G(x,n) are g.f.s of well-known sequences:
G(x,2) = g.f. of A000108 = 1 + x*G(x,2)^2;
G(x,3) = g.f. of A001764 = 1 + x*G(x,3)^3;
G(x,4) = g.f. of A002293 = 1 + x*G(x,4)^4 ; etc.
Explicitly, the product yielding the g.f. A(x) begins:
A(x) = [1 + x + x^2 + x^3 +...] * [1 + 2*x^2 + 5*x^4 + 14*x^6 +...] * [1 + 3*x^3 + 12*x^6 + 55*x^9 +...] * [1 + 4*x^4 + 22*x^8 + 140*x^12 +...] * ...

Crossrefs

Cf. A105862 (log(A(x))); A056045 (variant); A000108 (Catalan), A001764, A002293.

Programs

  • PARI
    a(n)=if(n==0,1, polcoeff(exp(sum(m=1,n, x^m*sumdiv(m,d, binomial(m,d)/gcd(m,d)))),n))
  • PARI
    a(n)=polcoeff(prod(m=1,n,(1/x*serreverse(x/(1+x^m +x*O(x^n))))^m),n)

Formula

G.f.: A(x) = exp( Sum_{n>=1} A105862(n)/n*x^n ), where A105862(n) = Sum_{d|n} binomial(n,d)*n/gcd(n,d).
G.f.: A(x) = Product_{n>=1} [ Series_Reversion( x/(1 + x^n) )/x ]^n.

A306843 a(n) = Sum_{d|n} binomial(n,d)^3.

Original entry on oeis.org

1, 9, 28, 281, 126, 11592, 344, 365465, 593434, 16095134, 1332, 921113624, 2198, 40424993884, 27175280778, 2137777203097, 4914, 121331143444050, 6860, 6310445825215406, 1572228697798262, 351047164202718608, 12168, 20174300460344963864, 149975199312626
Offset: 1

Views

Author

Seiichi Manyama, Mar 12 2019

Keywords

Links

Crossrefs

Sum_{d|n} binomial(n,d)^b: A000005 (b=0), A056045 (b=1), A174462 (b=2), this sequence (b=3).
Cf. A000172, A001093, A001158.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[n, #]^3 &]; Array[a, 25] (* Amiram Eldar, Jun 13 2021 *)
  • PARI
    {a(n) = sumdiv(n, d, binomial(n, d)^3)}

Formula

If p is prime, a(p) = A001093(p).

A329966 a(n) = n! * Sum_{d|n} binomial(n-1,d-1) / d!.

Original entry on oeis.org

1, 3, 7, 61, 121, 3721, 5041, 240241, 2056321, 23768641, 39916801, 11104853761, 6227020801, 683519316481, 32048919302401, 577844178912001, 355687428096001, 261396772808371201, 121645100408832001, 202418558674082150401, 2061884451929702400001, 12935940353987812761601
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2019

Keywords

Links

Crossrefs

Cf. A000262, A056045, A057625, A271654.

Programs

  • Magma
    [Factorial(n)*( &+[ Binomial(n-1,d-1)/Factorial(d):d in Divisors(n)]): n in [1..22]]; // Marius A. Burtea, Jan 02 2020
  • Maple
    N:= 30:
V:= Vector(N):
for d from 1 to N do
   for k from 1 to floor(N/d) do
     n:= k*d; V[n]:= V[n] + n!/d!*binomial(n-1,d-1);
od od:
convert(V,list); # Robert Israel, Jan 01 2020
  • Mathematica
    a[n_] := n! Sum[Binomial[n - 1, d - 1]/d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]
  • PARI
    a(n) = n! * sumdiv(n, d, binomial(n-1,d-1) / d!); \\ Michel Marcus, Nov 26 2019

A374770 a(n) is the number of subsets x of Z_n such that #x * #y = n and x + y = Z_n for some subset y of Z_n.

Original entry on oeis.org

1, 3, 4, 11, 6, 24, 8, 59, 40, 68, 12, 284, 14, 192, 384, 795, 18, 1590, 20, 2876, 2552, 2192, 24, 17972, 3156, 8388, 20560, 35620, 30, 116474, 32, 144091, 178512, 131396, 94968, 1118426, 38, 524688, 1596560, 2569884, 42, 7280934, 44
Offset: 1

Views

Author

Rémy Sigrist, Jul 19 2024

Keywords

Examples

			For n = 8: the principal subsets x (unique up to translation) alongside an appropriate subset y and the number of distinct translations are:
  x                  y                  #
  -----------------  -----------------  -
  {0}                {0,1,2,3,4,5,6,7}  8
  {0,1}              {0,2,4,6}          8
  {0,2}              {0,1,4,5}          8
  {0,3}              {0,2,4,6}          8
  {0,4}              {0,1,2,3}          4
  {0,1,2,3}          {0,4}              8
  {0,2,3,5}          {0,4}              8
  {0,1,4,5}          {0,2}              4
  {0,2,4,6}          {0,1}              2
  {0,1,2,3,4,5,6,7}  {0}                1
So a(8) = 8 + 8 + 8 + 8 + 4 + 8 + 8 + 4 + 2 + 1 = 59.

Links

Crossrefs

Cf. A045654, A056045, A374712.

Programs

  • Python
    from itertools import combinations
from sympy import divisors, isprime
def A374770(n):
    if isprime(n): return n+1
    s = {}
    for d in divisors(n,generator=True):
        t = {}
        for a in combinations(range(n),d):
            for i in range(1,n):
                if (w:=tuple((i+b)%n for b in a)) in t:
                    t[w]+=1
                    break
            else:
                t[a] = 1
        s[d] = t
    c = 0
    for d in divisors(n,generator=True):
        for a in s[d]:
            for b in s[n//d]:
                if len({(x+y)%n for x in a for y in b})==n:
                    c += s[d][a]
                    break
    return c # Chai Wah Wu, Jul 22 2024

Formula

a(p) = p + 1 for any prime number p.
a(n) <= A056045(n).

A082904 Triangle read by rows: Pascal's triangle restricted to binomial(n, d) where d is a divisor of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 6, 1, 5, 1, 6, 15, 20, 1, 7, 1, 8, 28, 70, 1, 9, 84, 1, 10, 45, 252, 1, 11, 1, 12, 66, 220, 495, 924, 1, 13, 1, 14, 91, 3432, 1, 15, 455, 3003, 1, 16, 120, 1820, 12870, 1, 17, 1, 18, 153, 816, 18564, 48620, 1, 19, 1, 20, 190, 4845, 15504, 184756, 1, 21, 1330
Offset: 0

Views

Author

Labos Elemer, Apr 23 2003

Keywords

Examples

			n-th row of table consists of A000005(n) terms:
   1;
   2,   1;
   3,   1;
   4,   6,   1;
   5,   1;
   6,  15,  20,   1;
   7,   1;
   8,  28,  70,   1;
   9,  84,   1;
  10,  45, 252,   1;
  ...
25th row = {C(25,1), C(25,5), C(25,25)} = {25, 53130, 1}.

Crossrefs

Cf. A000005, A007318, A056045.

Programs

  • Maple
    A082904_row := proc(n) seq(binomial(n,d),d=numtheory[divisors](n)) end:
seq(print(A082904_row(n)),n=1..10); # Peter Luschny, Dec 06 2011
  • Mathematica
    Flatten[Table[Binomial[n, Divisors[n]], {n, 1, 25}], 1]
  • PARI
    for(n=1,30,fordiv(n,d,print1(binomial(n,d)", "))) \\ Charles R Greathouse IV, Dec 06 2011

Extensions

New name from Peter Luschny, Dec 06 2011

A346188 a(1) = 1; a(n+1) = Sum_{d|n} binomial(n,d) * a(n/d).

Original entry on oeis.org

1, 1, 3, 10, 47, 236, 1482, 10375, 83351, 750412, 7506488, 82571369, 990876614, 12881395983, 180339682057, 2705095261250, 43281525456071, 735785932753208, 13244146802607336, 251638789249539385, 5032775785133933492, 105688291487814923233, 2325142412733663015287
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 09 2021

Keywords

Crossrefs

Cf. A000110, A003238, A056045, A330017, A345136.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, d] a[(n - 1)/d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 23}]

A367864 a(n) = Sum_{d|n} d * binomial(n,d).

Original entry on oeis.org

1, 4, 6, 20, 10, 102, 14, 352, 270, 1370, 22, 8340, 26, 24234, 16410, 110512, 34, 551754, 38, 1944880, 817992, 7760258, 46, 39190392, 265700, 135208502, 42190254, 570003392, 58, 2631501240, 62, 9701577536, 2128920354, 39671306930, 48694870, 179231802444, 74
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 03 2023

Keywords

Crossrefs

Cf. A056045, A271654.

Programs

  • Maple
    a:= n-> n*add(binomial(n-1, d-1), d=numtheory[divisors](n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 03 2023
  • Mathematica
    Table[Sum[d*Binomial[n, d], {d, Divisors[n]}], {n, 50}]
  • PARI
    a(n) = sumdiv(n, d, d * binomial(n,d)); \\ Michel Marcus, Dec 03 2023

Formula

a(p) = 2p, for p prime.
a(n) = n * A271654(n). - Alois P. Heinz, Dec 03 2023
Previous Showing 21-27 of 27 results.

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