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 1-10 of 12 results. Next

A173990 Incorrect version of A007178.

Original entry on oeis.org

1, 1, 3, 13, 75, 530, 4449, 43236
Offset: 1

Views

Author

Vladimir Dotsenko, Mar 04 2010

Keywords

Comments

Name was: Dimensions of components for the operad of level algebras.

A002572 Number of partitions of 1 into n powers of 1/2; or (according to one definition of "binary") the number of binary rooted trees.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 28, 50, 89, 159, 285, 510, 914, 1639, 2938, 5269, 9451, 16952, 30410, 54555, 97871, 175586, 315016, 565168, 1013976, 1819198, 3263875, 5855833, 10506175, 18849555, 33818794, 60675786, 108861148, 195312750, 350419594, 628704034, 1127987211, 2023774607, 3630948907
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

This is similar to a question about Egyptian fractions, except that there the denominators of the fractions must be distinct. - N. J. A. Sloane, Jan 13 2021
v(c, d) is the number of partitions of d into positive integers of the form d = c + c_1 + c_2 + ... + c_n, where c_1 <= 2*c, c_{i+1} <= 2*c_i. See Minc link.
Top row of Table 1 of Elsholtz. [Jonathan Vos Post, Aug 30 2011]
a(n+1) is the number of compositions n = p(1) + p(2) + ... + p(m) with p(1)=1 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]
Over an algebraically closed field of characteristic 2, a(n) gives dimensions of the generic cohomology groups H^i_gen(SL_2,L(1)) which are isomorphic to algebraic group cohomology groups H^i(SL_2,L(1)^[i]), where ^[i] denotes i-th Frobenius twist. [David I. Stewart, Oct 22 2013]

Examples

			{1}; {1/2 + 1/2}; { 1/2 + 1/4 + 1/4 }; { 1/2 + 1/4 + 1/8 + 1/8, 1/4 + 1/4 + 1/4 + 1/4 }; ...
From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(7+1)=16 compositions 7=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 2*p(k+1):
[ 1]  [ 1 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 1 2 ]
[ 3]  [ 1 1 1 1 2 1 ]
[ 4]  [ 1 1 1 2 1 1 ]
[ 5]  [ 1 1 1 2 2 ]
[ 6]  [ 1 1 2 1 1 1 ]
[ 7]  [ 1 1 2 1 2 ]
[ 8]  [ 1 1 2 2 1 ]
[ 9]  [ 1 1 2 3 ]
[10]  [ 1 2 1 1 1 1 ]
[11]  [ 1 2 1 1 2 ]
[12]  [ 1 2 1 2 1 ]
[13]  [ 1 2 2 1 1 ]
[14]  [ 1 2 2 2 ]
[15]  [ 1 2 3 1 ]
[16]  [ 1 2 4 ]
(End)
From _Joerg Arndt_, Dec 26 2012: (Start)
There are a(8)=16 partitions of 1 into 8 powers of 1/2 (dots denote zeros in the tables of multiplicities in the left column)
[ 1]  [ . 1 1 1 1 1 1 2 ]     + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 2/128
[ 2]  [ . 1 1 1 1 . 4 . ]     + 1/2 + 1/4 + 1/8 + 1/16 + 4/64
[ 3]  [ . 1 1 1 . 3 2 . ]     + 1/2 + 1/4 + 1/8 + 3/32 + 2/64
[ 4]  [ . 1 1 . 3 1 2 . ]     + 1/2 + 1/4 + 3/16 + 1/32 + 2/64
[ 5]  [ . 1 1 . 2 4 . . ]     + 1/2 + 1/4 + 2/16 + 4/32
[ 6]  [ . 1 . 3 1 1 2 . ]     + 1/2 + 3/8 + 1/16 + 1/32 + 2/64
[ 7]  [ . 1 . 3 . 4 . . ]     + 1/2 + 3/8 + 4/32
[ 8]  [ . 1 . 2 3 2 . . ]     + 1/2 + 2/8 + 3/16 + 2/32
[ 9]  [ . 1 . 1 6 . . . ]     + 1/2 + 1/8 + 6/16
[10]  [ . . 3 1 1 1 2 . ]     + 3/4 + 1/8 + 1/16 + 1/32 + 2/64
[11]  [ . . 3 1 . 4 . . ]     + 3/4 + 1/8 + 4/32
[12]  [ . . 3 . 3 2 . . ]     + 3/4 + 3/16 + 2/32
[13]  [ . . 2 3 1 2 . . ]     + 2/4 + 3/8 + 1/16 + 2/32
[14]  [ . . 2 2 4 . . . ]     + 2/4 + 2/8 + 4/16
[15]  [ . . 1 5 2 . . . ]     + 1/4 + 5/8 + 2/16
[16]  [ . . . 8 . . . . ]     + 8/8
(End)

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 192-194, 307.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002573, A002574, A007178, A047913, A049284, A049285, A102375, A294775.

Programs

  • Maple
    v := proc(c,d) option remember; local i; if d < 0 or c < 0 then 0 elif d = c then 1 else add(v(i,d-c),i=1..2*c); fi; end; [ seq(v(1,n), n=1..50) ];
  • Mathematica
    v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d - c], {i, 1, 2*c}]]]; a[n_] := v[1, n-1]; a[1] = 1; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Oct 19 2011, after Maple *)
  • PARI
    v(c,d) = if ( d<0 || c<0, 0, if ( d==c, 1, sum(i=1,2*c, v(i,d-c) ) ) )
  • PARI
    /* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
N=66;  q='q+O('q^N);
t=2;  /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503  */
L=2 + 2*ceil( log(N) / log(t) );
f(k)=(1-t^k)/(1-t);
la(j)=prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );
nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );
dn=sum(j=0, L, (-1)^j * la(j) );
gf=nm / dn;
Vec( gf )
/* Joerg Arndt, Dec 27 2012 */
  • PARI
    {a(n, k=2) = if( n<2 && k==2, n>=0, nMichael Somos, Dec 20 2016 */

Formula

From Jon E. Schoenfield, Dec 18 2016: (Start)
Numerically, it appears that
lim_{n->infinity} a(n)/c0^n = c1
and
lim_{n->infinity} (a(n)/c0^n - c1) / c2^n = c3
where
c0 = 1.79414718754168546349846498809380776370136441826513
55647141291458811011534167435879115275875728251544
55034381754309507738861994388752350104180891093803
27324310643547892073673907996758374498542252887021
90... = A102375
c1 = 0.14185320208540933707157739062733520381135377764439
00938624762999524081108574037129602775796177848175
96757823284956317508884467180902882086032012675483
68631687927534330190816399081295424373415296405657
19...
c2 = 0.71317957835995615685267138702642988919007297942329
35...
c3 = 0.06124104103121269745282188448763211918477582400104
06... (End)
a(n) = A294775(n-1,1). - Alois P. Heinz, Nov 08 2017

A294746 Number A(n,k) of compositions (ordered partitions) of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 10, 13, 1, 1, 1, 35, 217, 75, 1, 1, 1, 126, 4245, 8317, 525, 1, 1, 1, 462, 90376, 1239823, 487630, 4347, 1, 1, 1, 1716, 2019836, 216456376, 709097481, 40647178, 41245, 1, 1, 1, 6435, 46570140, 41175714454, 1303699790001, 701954099115, 4561368175, 441675, 1
Offset: 0

Views

Author

Alois P. Heinz, Nov 07 2017

Keywords

Comments

Row r >= 2, is asymptotic to r^(r*n + 3/2) / (2*Pi*n)^((r-1)/2). - Vaclav Kotesovec, Sep 20 2019

Examples

			A(3,1) = 13: [1/4,1/4,1/4,1/4], [1/2,1/4,1/8,1/8], [1/2,1/8,1/4,1/8], [1/2,1/8,1/8,1/4], [1/4,1/2,1/8,1/8], [1/4,1/8,1/2,1/8], [1/4,1/8,1/8,1/2], [1/8,1/2,1/4,1/8], [1/8,1/2,1/8,1/4], [1/8,1/4,1/2,1/8], [1/8,1/4,1/8,1/2], [1/8,1/8,1/2,1/4], [1/8,1/8,1/4,1/2].
Square array A(n,k) begins:
  1,   1,      1,         1,             1,                1, ...
  1,   1,      1,         1,             1,                1, ...
  1,   3,     10,        35,           126,              462, ...
  1,  13,    217,      4245,         90376,          2019836, ...
  1,  75,   8317,   1239823,     216456376,      41175714454, ...
  1, 525, 487630, 709097481, 1303699790001, 2713420774885145, ...

Links

Crossrefs

Columns k=0-10 give: A000012, A007178(n+1), A294850, A294851, A294852, A294853, A294854, A294855, A294856, A294857, A294858.
Rows n=0+1, 2-10 give: A000012, A001700, A294982, A294983, A294984, A294985, A294986, A294987, A294988, A294989.
Main diagonal gives: A294747.
Cf. A294775.

Programs

  • Maple
    b:= proc(n, r, p, k) option remember;
      `if`(n `if`(k=0, 1, b(k*n+1, 1, 0, k+1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
  • Mathematica
    b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, 0, k + 1]];
Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Formula

A(n,k) = [x^((k+1)^n)] (Sum_{j=0..k*n+1} x^((k+1)^j))^(k*n+1) for k>0, A(n,0) = 1.

A144690 Limit of the coefficient of x^(2^m+n) in B(x)^(n+1) as m grows, where B(x) = Sum_{k>=0} x^(2^k).

Original entry on oeis.org

1, 2, 6, 16, 130, 636, 5712, 34336, 811458, 7151380, 113034746, 1049982792, 25276020640, 293841338896, 5712436923000, 68827002466176, 3739997267623490, 60752008945662372, 1718332635327516238, 26832922324005759560, 1099199814287516279394
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2008

Keywords

Comments

The g.f. of A144691(n) = a(n)/(n+1) appears to have an interesting functional interpretation.
For a fixed n, the sequence of [x^(2^m+n)] B(x)^(n+1), m=0,1,2,... seems to stabilize at m = n + A023416(n). [From Max Alekseyev, Dec 19 2011]

Links

Crossrefs

Cf. A007178, A144691, A144692, A135068, A135069, A135070, A135071.

Programs

  • PARI
    { a(n) = local(m=n+log(n+.5)\log(2), B=sum(k=0,m,x^(2^k)));if(n<0, 0, polcoeff((B+O(x^(2^m+n+1)))^(n+1),2^m+n)) }

Formula

a(n) = (n+1)*A144691(n).

Extensions

a(14), a(15) corrected and a(16)-a(23) added by Max Alekseyev, May 03 2011
a(24)-a(27) in b-file from Max Alekseyev, Dec 19 2011

A144691 Limit of the coefficient of x^(2^m+n) in B(x)^(n+1)/(n+1) as m grows, where B(x) = Sum_{k>=0} x^(2^k).

Original entry on oeis.org

1, 1, 2, 4, 26, 106, 816, 4292, 90162, 715138, 10275886, 87498566, 1944309280, 20988667064, 380829128200, 4301687654136, 219999839271970, 3375111608092354, 90438559754079802, 1341646116200287978, 52342848299405537114, 921821277222438350170
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 26*x^4 + 106*x^5 + 816*x^6 +...
A(x/G(x)) = G(x) = x/Series_Reversion[x*A(x)], where
G(x) = 1 + x + x^2 + 17*x^4 + 408*x^6 + 69473*x^8 + 6018928*x^10 +...
and G(x) appears to continue with only even powers of x (cf. A144692).
The inverse binomial transform forms the g.f. of A202582:
A(x/(1+x))/(1+x) = 1 + x^2 + 19*x^4 + 515*x^6 + 74383*x^8 + 6816465*x^10 +...+ A202582(n)*x^n +...

Links

Crossrefs

Cf. A007178, A144690, A144692, A202582.

Programs

  • PARI
    { a(n) = local(m=n+log(n+.5)\log(2), B=sum(k=0,m,x^(2^k))); if(n<0, 0, polcoeff((B+O(x^(2^m+n+1)))^(n+1)/(n+1),2^m+n)) }

Formula

a(n) = A144690(n)/(n+1).
G.f. A(x) satisfies: A(x/(1+x))/(1+x) is an even function; i.e., the inverse binomial transform yields A202582.

Extensions

a(14), a(15) corrected and a(16)-a(23) added by Max Alekseyev, May 03 2011
a(24)-a(27) in b-file from Max Alekseyev, Dec 19 2011

A135068 a(n) = [x^(2^n+n-1)] (x + x^2 + x^4 + x^8 + ... + x^2^n)^n for n>=1.

Original entry on oeis.org

1, 2, 6, 16, 90, 636, 5712, 34336, 537282, 5941780, 99729146, 1049982792, 23200347040, 293841338896, 5712436923000, 68827002466176, 2844850573581890, 53069160498788772, 1545326270301621838, 26021954987946879560, 1020860369624228471394, 19905401189634441143740, 605270985059438427438138, 11141784565490367848976336, 621465511993167908247508400, 14470690420043042111787089216, 548187222712632324159956010732, 11881058908943228840652007583056
Offset: 1

Views

Author

Paul D. Hanna, Nov 17 2007

Keywords

Crossrefs

Cf. A007178 (variant); A135069, A135070 (variant), A135071.

Programs

  • Mathematica
    f[x_, n_] := (Sum[x^(2^k), {k, 0, n}])^n; Table[Coefficient[f[x, n], x^(2^n + n - 1)] , {n, 1, 20}] (* G. C. Greubel, Sep 22 2016 *)
  • PARI
    a(n)=if(n<1,0,polcoeff(sum(j=0,n,x^(2^j)+O(x^(2^n+n)))^n,2^n+n-1))

Formula

a(n) = A135069(n)*n.

Extensions

a(15)-a(19) from Alois P. Heinz, Apr 29 2009
a(20)-a(22) from Max Alekseyev, Dec 03 2010
a(23)-a(28) from Max Alekseyev, Aug 31 2024

A145073 Coefficient of x^(3^n) in Q(x)^(n+1), where Q(x) = Sum_{k>=0} (x^(3^k) + x^(2*3^k)).

Original entry on oeis.org

1, 2, 7, 44, 325, 3110, 34258, 436304, 6246439, 99637730, 1747851787, 33469239848, 694321125511, 15513919320368, 371408549270425, 9484764686963792, 257355204923817028, 7393783957451330630, 224224270472995407877
Offset: 0

Views

Author

Paul D. Hanna, Oct 09 2008

Keywords

Crossrefs

Cf. A007178, A145074, A145075, A145076.

Programs

  • PARI
    { a(n) = local( Q = sum(j=0,n,x^(3^j)+x^(2*3^j)) + O(x^(3^n+1)) ); polcoeff(Q^(n+1),3^n) }

Extensions

a(10)-a(12) from Alois P. Heinz, Apr 28 2009
a(13)-a(18) from Max Alekseyev, Dec 18 2011

A145074 Coefficient of x^(4^n) in Q(x)^(n+1) where Q(x) = Sum_{k>=0} (x^(4^k) + x^(2*4^k) + x^(3*4^k)).

Original entry on oeis.org

1, 3, 12, 97, 885, 10386, 141295, 2218987, 39171564, 770029605, 16652860169, 393052609026, 10051000032099, 276818387211471, 8168772682637220, 257133140827399753
Offset: 0

Views

Author

Paul D. Hanna, Oct 09 2008

Keywords

Crossrefs

Cf. A007178, A145073, A145075, A145076.

Programs

  • PARI
    {a(n,q=4)=local(Q=sum(j=0,n,(x^(q^j)-x^(q*q^j))/(1-x^(q^j)+O(x^(q^n+1)))));polcoeff(Q^(n+1),q^n)}

Extensions

a(8)-a(10) from Sean A. Irvine, Mar 31 2011
a(11)-a(15) from Max Alekseyev, Dec 19 2011

A145075 Coefficient of x^(5^n) in Q(x)^(n+1), where Q(x) = Sum_{k>=0} (x^(5^k) + x^(2*5^k) + x^(3*5^k) + x^(4*5^k)).

Original entry on oeis.org

1, 4, 18, 172, 1881, 25644, 411488, 7581792, 157296141, 3630709734, 92225760033, 2556393882912, 76774488792326, 2483261807354874
Offset: 0

Views

Author

Paul D. Hanna, Oct 09 2008

Keywords

Crossrefs

Cf. A007178, A145073, A145074, A145076.

Programs

  • PARI
    {a(n,q=5)=local(Q=sum(j=0,n,(x^(q^j)-x^(q*q^j))/(1-x^(q^j)+O(x^(q^n+1)))));polcoeff(Q^(n+1),q^n)}

Extensions

a(7)-a(13) from Max Alekseyev, Dec 19 2011

A145076 Coefficient of x^(6^n) in Q(x)^(n+1), where Q(x) = Sum_{k>=0} x^(6^k)*(1 - x^(5*6^k))/(1 - x^(6^k)).

Original entry on oeis.org

1, 5, 25, 270, 3430, 52996, 968905, 20342540, 480982030, 12646108250, 365943140101, 11555148366323, 395323293564108
Offset: 0

Views

Author

Paul D. Hanna, Oct 09 2008

Keywords

Crossrefs

Cf. A007178, A145073, A145074, A145075.

Programs

  • Maple
    Q:=proc(x,n) options operator, arrow: sum(x^(6^k)+x^(2*6^k)+x^(3*6^k)+x^(4*6^k)+x^(5*6^k),k=0..n) end proc: seq(coeff(Q(x,n)^(n+1),x,6^n),n=0..6); # Emeric Deutsch, Oct 20 2008
  • PARI
    {a(n,q=6)=local(Q=sum(j=0,n,(x^(q^j)-x^(q*q^j))/(1-x^(q^j)+O(x^(q^n+1)))));polcoeff(Q^(n+1),q^n)}

Extensions

a(6) from Emeric Deutsch, Oct 20 2008
a(7)-a(12) from Max Alekseyev, Dec 19 2011
Showing 1-10 of 12 results. Next

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