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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A093048 a(n) = n minus exponent of 2 in n, with a(0) = 0.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 5, 7, 5, 9, 9, 11, 10, 13, 13, 15, 12, 17, 17, 19, 18, 21, 21, 23, 21, 25, 25, 27, 26, 29, 29, 31, 27, 33, 33, 35, 34, 37, 37, 39, 37, 41, 41, 43, 42, 45, 45, 47, 44, 49, 49, 51, 50, 53, 53, 55, 53, 57, 57, 59, 58, 61, 61, 63, 58, 65, 65, 67, 66, 69
Offset: 0

Views

Author

Ralf Stephan, Mar 16 2004

Keywords

Examples

			G.f. = x + x^2 + 3*x^3 + 2*x^4 +  5*x^5 + 5*x^6 + 7*x^7 + 5*x^8 + 9*x^9 + ... - _Michael Somos_, Jan 25 2020

Links

Crossrefs

a(n) = n - A007814(n) = A093049(n) + 1, n > 0.
a(n) is the exponent of 2 in A002689(n-1), A014070(n), A060690(n), A075101(n).
See also A084623.

Programs

  • Maple
    A093048 := proc(n)
    n-A007814(n) ;
end proc: # R. J. Mathar, Jul 24 2014
  • Mathematica
    a[ n_] := If[ n == 0, n - IntegerExponent[n, 2]]; (* Michael Somos, Jan 25 2020 *)
  • PARI
    a(n) = if(n<1, 0, if(n%2==0, a(n/2) + n/2 - 1, n))
  • PARI
    a(n) = n - valuation(n, 2) \\ Jianing Song, Oct 24 2018
  • Python
    def A093048(n): return n-(~n& n-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Formula

Recurrence: a(2n) = a(n) + n - 1, a(2n+1) = 2n + 1.
G.f.: Sum_{k>=0} (t*(t^3 + t^2 + 1)/(1 - t^2)^2), with t = x^2^k.
a(n) = Sum_{k=1..n} sign(n mod 2^k). - Wesley Ivan Hurt, May 09 2021

A166995 G.f.: C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 0, 8, 32, 2848, 87808, 97425920, 18364346368, 459757145081856, 468713931103109120, 349620381018764380930048, 1788712998645738038832398336, 46562065744123901943395531497144320
Offset: 0

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Examples

			G.f: C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
The g.f. of A166996 is S(x):
S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!
S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...

Links

Crossrefs

Cf. A166996, A166997, A166998, A060690, A014070.

Programs

  • Mathematica
    Table[(1/2)*(Binomial[2^n + n - 1, n ] + (-1)^n *Binomial[2^n, n]), {n, 0, 10}] (* G. C. Greubel, May 30 2016 *)
  • PARI
    {a(n)=polcoeff(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!),n)}
  • PARI
    {a(n)=(binomial(2^n + n-1, n) + (-1)^n*binomial(2^n, n))/2} \\ Paul D. Hanna, Nov 24 2009

Formula

a(n) = ( C(2^n + n-1, n) + (-1)^n*C(2^n, n) )/2. - Paul D. Hanna, Nov 24 2009

A166996 G.f.: S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!, a power series in x with integer coefficients.

Original entry on oeis.org

2, 2, 88, 1028, 289184, 22451552, 112890141568, 50093449805856, 6676830881369059840, 15354513520142235310592, 66620888067382334066280699904, 750203718611121304644623635491840
Offset: 1

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Examples

			G.f.: S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 + ...
The g.f. of A166995 is C(x):
C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!.
C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 + ...
where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 + ...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 + ...

Links

Crossrefs

Cf. A166995, A166997, A166998, A060690, A014070.

Programs

  • Mathematica
    Table[(1/2)*(Binomial[2^n + n - 1, n ] - (-1)^n *Binomial[2^n, n]), {n, 50}] (* G. C. Greubel, May 30 2016 *)
  • PARI
    {a(n)=polcoeff(-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!),n)}
  • PARI
    {a(n)=(binomial(2^n + n-1, n) - (-1)^n*binomial(2^n, n))/2} \\ Paul D. Hanna, Nov 24 2009

Formula

a(n) = (binomial(2^n + n-1, n) - (-1)^n*binomial(2^n, n) )/2. [Paul D. Hanna, Nov 24 2009]

A166998 G.f.: sqrt(C(x)^2 - S(x)^2) where C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.

Original entry on oeis.org

1, 0, 6, 28, 2684, 85664, 96848424, 18318978896, 459531493100736, 468613553577122688, 349607028167776160389536, 1788682277200384090414421312, 46561932503015793339090359576558496
Offset: 0

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Examples

			G.f: 1 + 6*x^2 + 28*x^3 + 2684*x^4 + 85664*x^5 + 96848424*x^6 +...
which equals sqrt( C(x)^2 - S(x)^2 ) where
C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...

Crossrefs

Cf. A166995, A166996, A166997, A060690, A014070.

Programs

  • PARI
    {a(n)=polcoeff(sqrt(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!)^2-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!)^2),n)}

Formula

G.f.: sqrt([C(x)+S(x)]*[C(x)-S(x)]) where C(x) + S(x) = g.f. of A060690 and C(-x) - S(-x) = g.f. of A014070.
Self-convolution yields A166998.

A220886 Irregular triangular array read by rows: T(n,k) is the number of inequivalent n X n {0,1} matrices modulo permutation of the rows, containing exactly k 1's; n>=0, 0<=k<=n^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 9, 20, 27, 27, 20, 9, 3, 1, 1, 4, 16, 48, 133, 272, 468, 636, 720, 636, 468, 272, 133, 48, 16, 4, 1, 1, 5, 25, 95, 330, 1027, 2780, 6550, 13375, 23700, 36403, 48405, 55800, 55800, 48405, 36403, 23700, 13375, 6550, 2780, 1027, 330, 95, 25, 5, 1
Offset: 0

Views

Author

Geoffrey Critzer, Feb 20 2013

Keywords

Comments

In other words, two matrices are considered equivalent if one can be obtained from the other by some sequence of interchanges of the rows.

Examples

			T(2,2) = 4 because we have: {{0,0},{1,1}}; {{0,1},{1,0}}; {{0,1},{0,1}}; {{1,0},{1,0}} (where the first two matrices were arbitrarily selected as class representatives).
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  4,  2,   1;
  1, 3,  9, 20,  27,  27,  20,   9,   3,   1;
  1, 4, 16, 48, 133, 272, 468, 636, 720, 636, 468, 272, 133, 48, 16, 4, 1;
  ...

Links

Crossrefs

Row sums are A060690.
Columns k=0-3 give: A000012, A000027, A000290 (n>=2), A203552 (n>=3).
Main diagonal gives A360660.
Cf. A360693.

Programs

  • Maple
    g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$3)):
seq(T(n), n=0..5);  # Alois P. Heinz, Feb 15 2023
  • Mathematica
    nn=100;Table[CoefficientList[Series[CycleIndex[SymmetricGroup[n],s]/.Table[s[i]->(1+x^i)^n,{i,1,n}],{x,0,nn}],x],{n,0,5}]//Grid
(* Second program: *)
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i] + k - 1, k], {k, 0, j}]]]];
T[n_] := CoefficientList[g[n, n, n], x];
Table[T[n], {n, 0, 5}] // Flatten (* Jean-François Alcover, May 28 2023, after Alois P. Heinz *)

A060336 Number of n X n {-1,0,1} matrices modulo rows permutation (by symmetry this is the same as the number of {-1,0,1} matrices modulo columns permutation), i.e., the number of equivalence classes where two matrices A and B are equivalent if one of them is the result of permuting the rows of the other.

Original entry on oeis.org

3, 45, 3654, 1929501, 7355513529, 212787633478239, 47937678641708357304, 85524882506287709213421693, 1224201212028616655577478516173315, 142132497715474639139076246298436794277130
Offset: 1

Views

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001

Keywords

Links

Crossrefs

A060690.

Programs

  • Mathematica
    Table[Binomial[3^n+n-1,n],{n,10}] (* Harvey P. Dale, Apr 10 2012 *)
  • PARI
    { for (n=1, 47, write("b060336.txt", n, " ", binomial(3^n + n - 1, n)); ) } \\ Harry J. Smith, Jul 03 2009

Formula

a(n) = C(3^n + n - 1, n) (where C(n, k) denotes the binomial coefficient).
a(n) ~ 3^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

Extensions

More terms from Harry J. Smith, Jul 03 2009

A156631 G.f.: A(x) = Sum_{n>=0} ( Sum_{k>=1} (2^n*2^k*x)^k/k )^n / n!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 64, 3072, 466944, 283115520, 814634500096, 10734635101192192, 601470215201514061824, 138785509787119430915850240, 130376354694095237162362352959488
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2009

Keywords

Comments

Compare to these dual g.f.s:
Sum_{n>=0} ( Sum_{k>=1} (2^n*x)^k/k )^n/n! (A060690);
Sum_{n>=0} ( Sum_{k>=1} (2^k*x)^k/k )^n/n! (A155200);
which, when expanded as power series in x, have only integer coefficients.

Examples

			G.f.: A(x) = 1 + 4*x + 64*x^2 + 3072*x^3 + 466944*x^4 + 283115520*x^5 + ...
From _Paul D. Hanna_, Mar 10 2009: (Start)
Let B(x) be the g.f. of A155200:
B(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 + ...
then a(n) is the coefficient of x^n in B(x)^(2^n):
B(x)^(2^0): [(1),2,10,188,16774,6745436,11466849412,...];
B(x)^(2^1): [1,(4),24,416,34400,13561728,22961051392,...];
B(x)^(2^2): [1,8,(64),1024,72704,27418624,46032420864,...];
B(x)^(2^3): [1,16,192,(3072),165888,56131584,92513894400,...];
B(x)^(2^4): [1,32,640,12288,(466944),118751232,186897137664,...];
B(x)^(2^5): [1,64,2304,65536,2129920,(283115520),382143037440,...];
B(x)^(2^6): [1,128,8704,425984,17956864,1140850688,(814634500096),...];
the terms along the diagonal (in parentheses) form this sequence. (End)

Crossrefs

Cf. A156630, A060690, A155200.

Programs

  • PARI
    {a(n)=polcoeff(sum(j=0,n,sum(k=1, n, (2^(j+k)*x)^k/k+x*O(x^n))^j/j!),n)}
  • PARI
    /* a(n) = [x^n] B(x)^(2^n) where B(x) is g.f. of A155200: */ {a(n)=polcoeff(exp( 2^n*sum(k=1,n, 2^(k^2)*x^k/k)+x*O(x^n)), n)} \\ Paul D. Hanna, Mar 11 2009

Formula

a(n) = [x^n] B(x)^(2^n) where B(x) = exp(Sum_{n>=1} 2^(n^2)*x^n/n) is the g.f. of A155200. - Paul D. Hanna, Mar 10 2009

A166997 G.f.: C(x)^2 - S(x)^2 where C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.

Original entry on oeis.org

1, 0, 12, 56, 5404, 171664, 193729840, 36639136064, 919064160383600, 937227332865348224, 699214061851483321467008, 3577364560049979516493456896, 93123865010226899737836259608990464
Offset: 0

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Examples

			G.f: 1 + 12*x^2 + 56*x^3 + 5404*x^4 + 171664*x^5 + 193729840*x^6 +...
which equals C(x)^2 - S(x)^2 where
C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...

Crossrefs

Cf. A166995, A166996, A166998, A060690, A014070.

Programs

  • PARI
    {a(n)=polcoeff(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!)^2-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!)^2,n)}

Formula

G.f.: [C(x)+S(x)]*[C(x)-S(x)] where C(x) + S(x) = g.f. of A060690 and C(-x) - S(-x) = g.f. of A014070.
Self-convolution of A166998.

A180687 G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x/(1-x))^n/n!.

Original entry on oeis.org

1, 2, 8, 70, 2008, 209018, 76000040, 94978699326, 410326957417208, 6211396910763188786, 334321755307017208207432, 64835518006826024523658441206, 45812575197824183928260946747286552
Offset: 0

Views

Author

Paul D. Hanna, Sep 16 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 70*x^4 + 2008*x^5 +...
A(x) = Sum_{n>=0} log(1 + 2^n*x + 2^n*x^2 + 2^n*x^3 + 2^n*x^4 +...)^n/n!.
A(x) = 1 + log(1+2x/(1-x)) + log(1+4x/(1-x))^2/2! + log(1+8x/(1-x))^3/3! +...

Crossrefs

Cf. variants: A159602, A060690.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=0, n, log(1+2^m*x/(1-x+x*O(x^n)))^m/m!), n)}

Formula

a(n) = Sum_{k=0..n} binomial(2^k, k) * binomial(n-1, n-k) for n >= 0. - Paul D. Hanna, Apr 04 2023

A165984 Number of ways to put n indistinguishable balls into n^3 distinguishable boxes.

Original entry on oeis.org

1, 1, 36, 3654, 766480, 275234400, 151111164204, 117774526188844, 123672890985095232, 168324948170849366820, 288216356245328994082600, 606320062786763763996747618, 1537230010624231669678572481296, 4622745700243196227504110670860680
Offset: 0

Views

Author

Thomas Wieder, Oct 03 2009

Keywords

Comments

See A165817 for the case n indistinguishable balls into 2*n distinguishable boxes.
See A054688 for the case n indistinguishable balls into n^2 distinguishable boxes.
a(n) is the number of (weak) compositions of n into n^3 parts. - Joerg Arndt, Oct 04 2017

Examples

			For n = 2 the a(2) = 36 solutions are
[0, 0, 0, 0, 0, 0, 0, 2]
[0, 0, 0, 0, 0, 0, 1, 1]
[0, 0, 0, 0, 0, 0, 2, 0]
[0, 0, 0, 0, 0, 1, 0, 1]
[0, 0, 0, 0, 0, 1, 1, 0]
[0, 0, 0, 0, 0, 2, 0, 0]
[0, 0, 0, 0, 1, 0, 0, 1]
[0, 0, 0, 0, 1, 0, 1, 0]
[0, 0, 0, 0, 1, 1, 0, 0]
[0, 0, 0, 0, 2, 0, 0, 0]
[0, 0, 0, 1, 0, 0, 0, 1]
[0, 0, 0, 1, 0, 0, 1, 0]
[0, 0, 0, 1, 0, 1, 0, 0]
[0, 0, 0, 1, 1, 0, 0, 0]
[0, 0, 0, 2, 0, 0, 0, 0]
[0, 0, 1, 0, 0, 0, 0, 1]
[0, 0, 1, 0, 0, 0, 1, 0]
[0, 0, 1, 0, 0, 1, 0, 0]
[0, 0, 1, 0, 1, 0, 0, 0]
[0, 0, 1, 1, 0, 0, 0, 0]
[0, 0, 2, 0, 0, 0, 0, 0]
[0, 1, 0, 0, 0, 0, 0, 1]
[0, 1, 0, 0, 0, 0, 1, 0]
[0, 1, 0, 0, 0, 1, 0, 0]
[0, 1, 0, 0, 1, 0, 0, 0]
[0, 1, 0, 1, 0, 0, 0, 0]
[0, 1, 1, 0, 0, 0, 0, 0]
[0, 2, 0, 0, 0, 0, 0, 0]
[1, 0, 0, 0, 0, 0, 0, 1]
[1, 0, 0, 0, 0, 0, 1, 0]
[1, 0, 0, 0, 0, 1, 0, 0]
[1, 0, 0, 0, 1, 0, 0, 0]
[1, 0, 0, 1, 0, 0, 0, 0]
[1, 0, 1, 0, 0, 0, 0, 0]
[1, 1, 0, 0, 0, 0, 0, 0]
[2, 0, 0, 0, 0, 0, 0, 0]

Crossrefs

Cf. A001700, A054688, A060690, A165817.

Programs

  • Maple
    a:= n-> binomial(n^3+n-1, n): seq(a(n), n=0..16);
  • Mathematica
    Table[Binomial[n^3 + n - 1, n], {n, 0, 13}] (* Michael De Vlieger, Oct 05 2017 *)
  • PARI
    a(n) = binomial(n^3+n-1, n); \\ Altug Alkan, Oct 03 2017

Formula

a(n) = binomial(n^3+n-1, n).
Let denote P(n) = the number of integer partitions of n,
p(i) = the number of parts of the i-th partition of n,
d(i) = the number of different parts of the i-th partition of n,
m(i,j) = multiplicity of the j-th part of the i-th partition of n.
Then one has:
a(n) = Sum_{i=1..P(n)} (n^3)!/((n^3-p(i))!*(Product_{j=1..d(i)} m(i,j)!)).
a(n) = [x^n] 1/(1 - x)^(n^3). - Ilya Gutkovskiy, Oct 03 2017
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