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.

Showing 1-6 of 6 results.

A070933 Expansion of Product_{k>=1} 1/(1 - 2*t^k).

Original entry on oeis.org

1, 2, 6, 14, 34, 74, 166, 350, 746, 1546, 3206, 6550, 13386, 27114, 54894, 110630, 222794, 447538, 898574, 1801590, 3610930, 7231858, 14480654, 28983246, 58003250, 116054034, 232186518, 464475166, 929116402, 1858449178, 3717247638, 7434950062, 14870628026, 29742206138, 59485920374, 118973809798, 237950730522, 475905520474
Offset: 0

Views

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 21 2002

Keywords

Comments

See A083355 for a similar formula. - Thomas Wieder, May 07 2008
Partitions of n into 2 sorts of parts: the parts are unordered, but not the sorts; see example and formula by Wieder. - Joerg Arndt, Apr 28 2013
Convolution inverse of A070877. - George Beck, Dec 02 2018
Number of conjugacy classes of n X n matrices over GF(2). Cf. Morrison link, section 2.9. - Geoffrey Critzer, May 26 2021

Examples

			From _Joerg Arndt_, Apr 28 2013: (Start)
There are a(3)=14 partitions of 3 with 2 ordered sorts. Here p:s stands for "part p of sort s":
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:1  1:0  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:1  1:0  1:0  ]
06:  [ 1:1  1:0  1:1  ]
07:  [ 1:1  1:1  1:0  ]
08:  [ 1:1  1:1  1:1  ]
09:  [ 2:0  1:0  ]
10:  [ 2:0  1:1  ]
11:  [ 2:1  1:0  ]
12:  [ 2:1  1:1  ]
13:  [ 3:0  ]
14:  [ 3:1  ]
(End)

Links

Crossrefs

Cf. A006951, A000041, A070877.
Cf. A083355.
Column k=2 of A246935.
Cf. A048651.
Row sums of A256193.
Antidiagonal sums of A322210.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-2*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018
  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 07 2014
  • Mathematica
    CoefficientList[ Series[ Product[1 / (1 - 2t^k), {k, 1, 35}], {t, 0, 35}], t]
CoefficientList[Series[E^Sum[2^k*x^k / (k*(1-x^k)), {k,1,30}],{x,0,30}],x] (* Vaclav Kotesovec, Sep 09 2014 *)
(O[x]^20 - 1/QPochhammer[2,x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)
  • Maxima
    S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */
  • PARI
    N=66; q='q+O('q^N); Vec(1/sum(n=0, N, (-2)^n*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ Joerg Arndt, Mar 09 2014

Formula

a(n) = (1/n)*Sum_{k=1..n} A054598(k)*a(n-k). - Vladeta Jovovic, Nov 23 2002
a(n) is asymptotic to c*2^n where c=3.46253527447396564949732... - Benoit Cloitre, Oct 26 2003. Right value of this constant is c = 1/A048651 = 3.46274661945506361153795734292443116454075790290443839132935303175891543974042... . - Vaclav Kotesovec, Sep 09 2014
Euler transform of A000031(n). - Vladeta Jovovic, Jun 23 2004
a(n) = Sum_{k=1..n} p(n,k)*A000079(k) where p(n,k) = number of integer partitions of n into k parts. - Thomas Wieder, May 07 2008
a(n) = S(n,1), where S(n,m) = 2 + Sum_{k=m..floor(n/2)} 2*S(n-k,k), S(n,n)=2, S(0,m)=1, S(n,m)=0 for n < m. - Vladimir Kruchinin, Sep 07 2014
a(n) = Sum_{lambda,mu,nu} (c^{lambda}{mu,nu})^2, where lambda ranges over all partitions of n, mu and nu range over all partitions satisfying |mu| + |nu| = n, and c^{lambda}{mu,nu} denotes a Littlewood-Richardson coefficient. - Richard Stanley, Nov 16 2014
G.f.: Sum_{i>=0} 2^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018
G.f.: Product_{j>=1} Product_{i>=1} 1/(1-x^(i*j))^A001037(j) given in Morrison link section 2.9. - Geoffrey Critzer, May 26 2021

Extensions

Edited and extended by Robert G. Wilson v, May 25 2002

A322200 L.g.f.: L(x,y) = log( Product_{n>=1} 1/(1 - (x^n + y^n)) ), where L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k / (n+k) such that L(0,0) = 0, as a symmetric square table of coefficients T(n,k) read by antidiagonals starting with T(0,0) = 0.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 4, 3, 3, 4, 7, 4, 10, 4, 7, 6, 5, 10, 10, 5, 6, 12, 6, 21, 26, 21, 6, 12, 8, 7, 21, 35, 35, 21, 7, 8, 15, 8, 36, 56, 90, 56, 36, 8, 15, 13, 9, 36, 93, 126, 126, 93, 36, 9, 13, 18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28, 14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14, 24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24, 24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24, 31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2018

Keywords

Examples

			L.g.f.: L(x,y) = (x + y)/1 + (3*x^2 + 2*x*y + 3*y^2)/2 + (4*x^3 + 3*x^2*y + 3*x*y^2 + 4*y^3)/3 + (7*x^4 + 4*x^3*y + 10*x^2*y^2 + 4*x*y^3 + 7*y^4)/4 + (6*x^5 + 5*x^4*y + 10*x^3*y^2 + 10*x^2*y^3 + 5*x*y^4 + 6*y^5)/5 + (12*x^6 + 6*x^5*y + 21*x^4*y^2 + 26*x^3*y^3 + 21*x^2*y^4 + 6*x*y^5 + 12*y^6)/6 + (8*x^7 + 7*x^6*y + 21*x^5*y^2 + 35*x^4*y^3 + 35*x^3*y^4 + 21*x^2*y^5 + 7*x*y^6 + 8*y^7)/7 + (15*x^8 + 8*x^7*y + 36*x^6*y^2 + 56*x^5*y^3 + 90*x^4*y^4 + 56*x^3*y^5 + 36*x^2*y^6 + 8*x*y^7 + 15*y^8)/8 + ...
such that
exp( L(x,y) ) = Product_{n>=1} 1/(1 - (x^n + y^n)), or
L(x,y) = Sum_{n>=1} -log(1 - (x^n + y^n)),
where
L(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n+k),
in which the constant term is taken to be zero: L(0,0) = 0.
SQUARE TABLE.
The square table of coefficients T(n,k) of x^n*y^k/(n+k) in L(x,y) begins
0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...;
3, 3, 10, 10, 21, 21, 36, 36, 55, 55, 78, 78, 105, ...;
4, 4, 10, 26, 35, 56, 93, 120, 165, 232, 286, 364, ...;
7, 5, 21, 35, 90, 126, 230, 330, 537, 715, 1043, 1365, ...;
6, 6, 21, 56, 126, 262, 462, 792, 1287, 2002, 3018, ...;
12, 7, 36, 93, 230, 462, 994, 1716, 3073, 5035, 8120, ...;
8, 8, 36, 120, 330, 792, 1716, 3446, 6435, 11440, 19448, ...;
15, 9, 55, 165, 537, 1287, 3073, 6435, 13050, 24310, 44010, ...;
13, 10, 55, 232, 715, 2002, 5035, 11440, 24310, 48698, 92378, ...;
18, 11, 78, 286, 1043, 3018, 8120, 19448, 44010, 92378, 185310, ...;
12, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, ...; ...
TRIANGLE.
Alternatively, this sequence may be written as a triangle, starting as
0;
1, 1;
3, 2, 3;
4, 3, 3, 4;
7, 4, 10, 4, 7;
6, 5, 10, 10, 5, 6;
12, 6, 21, 26, 21, 6, 12;
8, 7, 21, 35, 35, 21, 7, 8;
15, 8, 36, 56, 90, 56, 36, 8, 15;
13, 9, 36, 93, 126, 126, 93, 36, 9, 13;
18, 10, 55, 120, 230, 262, 230, 120, 55, 10, 18;
12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12;
28, 12, 78, 232, 537, 792, 994, 792, 537, 232, 78, 12, 28;
14, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 14;
24, 14, 105, 364, 1043, 2002, 3073, 3446, 3073, 2002, 1043, 364, 105, 14, 24;
24, 15, 105, 470, 1365, 3018, 5035, 6435, 6435, 5035, 3018, 1365, 470, 105, 15, 24;
31, 16, 136, 560, 1892, 4368, 8120, 11440, 13050, 11440, 8120, 4368, 1892, 560, 136, 16, 31; ...
where L(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n-k,k)*x^(n-k)*y^k / n.

Links

Crossrefs

Cf. A322210 (exp), A322201 (main diagonal), A322203, A322205, A322207, A322209.
Cf. A054598 (antidiagonal sums), A054599.

Programs

  • PARI
    {L = sum(n=1,61, -log(1 - (x^n + y^n) +O(x^61) +O(y^61)) );}
{T(n,k) = polcoeff( (n+k)*polcoeff( L,n,x),k,y)}
for(n=0,16, for(k=0,16, print1( T(n,k),", ") );print(""))

Formula

Sum_{k=0..n} T(n-k,k) = A054598(n) = Sum_{d|n} d*2^(n/d).
Sum_{k=0..n} T(n-k,k) * k/n = A054599(n) = Sum_{d|n} d*2^(n/d - 1).
Sum_{k=0..n} T(n-k,k) * 2^k = A322209(n) = [x^n] log( Product_{k>=1} 1/(1 - (2^k+1)*x^k) ) for n >= 0.
FORMULAS FOR TERMS.
T(n,k) = T(k,n) for n >= 0, k >= 0.
T(0,0) = 0.
T(n,0) = sigma(n) for n > 0.
T(0,k) = sigma(k) for n > 0.
T(n,1) = n+1, for n >= 0.
T(1,k) = k+1, for k >= 0.
T(2*n,2) = T(2*n+1,2) = (n+1)*(2*n+3).
T(2,2*k) = T(2,2*k+1) = (k+1)*(2*k+3).
COLUMN GENERATING FUNCTIONS.
Row 0: log(P(x)), where P(x) = Product_{n>=1} 1/(1 - x^n).
Row 1: 1/(1-x)^2.
Row 2: (3 + x^2)/((1-x)*(1-x^2)^2).
Row 3: (4 - 4*x + 6*x^2 + 2*x^3 + x^4)/((1-x)^2*(1-x^3)^2).
Row 4: (7 - 9*x + 11*x^2 + 7*x^3 + 9*x^4 + x^5 + 5*x^6 + x^7)/((1-x)^2*(1-x^2)*(1-x^4)^2).
Row 5: (6 - 18*x + 33*x^2 - 16*x^3 + 10*x^4 + 4*x^5 + 3*x^6 + 2*x^7 + x^8)/((1-x)^3*(1-x^5)^2).
Row 6: (12 - 41*x + 56*x^2 + 13*x^3 - 49*x^4 - 20*x^5 + 105*x^6 - 126*x^7 + 85*x^8 - 62*x^9 + 24*x^10 - 28*x^11 + 39*x^12 - 25*x^13 + 15*x^14 + x^15 + x^16) / ((1-x)^4*(1-x^2)^2*(1-x^3)*(1-x^6)^2).

A054599 a(n) = Sum_{d|n} d*2^(n/d - 1).

Original entry on oeis.org

0, 1, 4, 7, 16, 21, 52, 71, 160, 277, 564, 1035, 2176, 4109, 8348, 16467, 33088, 65553, 131740, 262163, 525456, 1048817, 2099244, 4194327, 8393344, 16777321, 33562676, 67109695, 134234480, 268435485, 536905572, 1073741855, 2147549824
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Examples

			G.f. = x + 4*x^2 + 7*x^3 + 16*x^4 + 21*x^5 + 52*x^6 + 71*x^7 + 160*x^8 + 277*x^9 + ...

Links

Crossrefs

Cf. A000016, A000031, A054598, A054600, A054601.

Programs

  • Mathematica
    {0}~Join~Table[DivisorSum[n, 2^(n/# - 1) # &], {n, 1, 20}] (* Vladimir Reshetnikov, Nov 20 2015 *)
Table[SeriesCoefficient[-Log[-QPochhammer[2, x]] n/2, {x, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 20 2015 *)
  • PARI
    a(n) = if (n<1, 0, sumdiv(n, d, d*2^(n/d - 1))); \\ Michel Marcus, Nov 21 2015

Formula

G.f.: Sum_{n>0} n*x^n/(1-2*x^n). - Vladeta Jovovic, Oct 27 2002
L.g.f.: -log(-(2;-x)inf)/2, where (a;q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2015
G.f.: Sum_{k>=1} 2^(k-1)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Sep 10 2019
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Oct 16 2019

A179469 G.f. satisfies A(x) = exp( Sum_{n>=1} 2^n*A(x^n)*x^n/n ).

Original entry on oeis.org

1, 2, 8, 32, 140, 624, 2928, 14048, 69200, 347040, 1768120, 9122144, 47572128, 250341312, 1327718272, 7089595552, 38082093120, 205638343552, 1115635692576, 6078058719232, 33239328613648, 182402290944576, 1004073853702320
Offset: 0

Views

Author

Paul D. Hanna, Jul 15 2010

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 32*x^3 + 140*x^4 + 624*x^5 + +...
log(A(x)) = 2*A(x) + 4*A(x^2)*x^2/2 + 8*A(x^3)*x^3/3 + 16*A(x^4)*x^4/4 +...

Links

Crossrefs

Cf. A054598, A179470.

Programs

  • PARI
    {a(n)=my(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A,x,x^m+x*O(x^n))*2^m*x^m/m)));polcoeff(A,n)}

Formula

From Seiichi Manyama, Jun 02 2023: (Start)
A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-2*x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 2^(k/d) * a(d-1) ) * a(n-k). (End)

A054601 a(n) = Sum_{d|n, d odd} d*2^(n/d - 1), a(0)=0.

Original entry on oeis.org

0, 1, 2, 7, 8, 21, 38, 71, 128, 277, 522, 1035, 2072, 4109, 8206, 16467, 32768, 65553, 131186, 262163, 524328, 1048817, 2097174, 4194327, 8388992, 16777321, 33554458, 67109695, 134217784, 268435485, 536872638, 1073741855, 2147483648
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Links

Crossrefs

Cf. A054598, A054599, A054600, A000016, A000031.

Programs

  • PARI
    a(n) = if (n, sumdiv(n, d, if (d%2, d*2^(n/d - 1))), 0); \\ Michel Marcus, Mar 10 2021

Formula

G.f.: Sum_{k>0} (1-(-1)^k)/2*k*x^k/(1-2*x^k). - Vladeta Jovovic, Oct 17 2003

A054600 Sum_{d|n, d odd} d*2^(n/d).

Original entry on oeis.org

0, 2, 4, 14, 16, 42, 76, 142, 256, 554, 1044, 2070, 4144, 8218, 16412, 32934, 65536, 131106, 262372, 524326, 1048656, 2097634, 4194348, 8388654, 16777984, 33554642, 67108916, 134219390, 268435568, 536870970, 1073745276, 2147483710
Offset: 0

Views

Author

N. J. A. Sloane, Apr 16 2000

Keywords

Crossrefs

Cf. A054598-A054601, A000016, A000031.

Programs

  • Mathematica
    f[n_]:=With[{od=Select[Divisors[n],OddQ]},Total[od 2^(n/od)]]; Join[{0}, Array[f, 40]] (* Harvey P. Dale, Aug 31 2024 *)
Showing 1-6 of 6 results.

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