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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A134040 a(0) = 1; for n>0, a(n) = number of binary partitions of the Catalan number A000108(n).

Original entry on oeis.org

1, 2, 4, 14, 140, 4964, 808870, 726210606, 4161522164020, 170403742275382924, 54674613696351170731038, 148019646825727958873435181692, 3596203368022579371689526442266893534
Offset: 0

Views

Author

Paul D. Hanna, Oct 02 2007

Keywords

Examples

			a(0)=1, a(1)=A000123(1)=2, a(2)=A000123(2)=4, a(3)=A000123(5)=14, a(4)=A000123(14)=140.

Links

Crossrefs

Cf. A000123, A000108.

Formula

a(n) = A000123( A000108(n) ) for n>0 with a(0)=1, where A000108(n) = C(2*n,n)/(n+1) (Catalan numbers) and A000123(n) = number of partitions of 2n into powers of 2.

A134041 a(n) = number of binary partitions of the Fibonacci number F(n).

Original entry on oeis.org

1, 2, 2, 4, 6, 14, 36, 114, 450, 2268, 14442, 118686, 1264678, 17519842, 318273566, 7607402556, 240151303078, 10055927801538, 559859566727028, 41582482495661986, 4129785050606801246, 549628445573614296188, 98256218721544814784486, 23631541930531250077261282
Offset: 0

Views

Author

Paul D. Hanna, Oct 02 2007

Keywords

Links

Crossrefs

Cf. A000123, A000045.

Programs

  • Maple
    g:= proc(b, n) option remember; local t; if b<0 then 0 elif b=0 or n=0 then 1 elif b>=n then add(g(b-t, n) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1) else g(b-1, n) +g(2*b, n-1) fi end: f:= proc(n) local t; t:= ilog2(2*n+1); g(n/2^(t-1), t) end: a:= n-> f(combinat[fibonacci](n)): seq(a(n), n=0..25);  # Alois P. Heinz, Sep 26 2011
  • Mathematica
    g[b_, n_] := g[b, n] = If[b < 0, 0, If[b == 0 || n == 0, 1, If[b >= n, Sum[g[b - t, n] Binomial[n + 1, t] (-1)^(t + 1), {t, 1, n + 1}], g[b - 1, n] + g[2b, n - 1]]]];
f[n_] := With[{t = Floor@Log[2, 2n + 1]}, g[n/2^(t - 1), t]];
a[n_] := f[Fibonacci[n]];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 19 2020, after Alois P. Heinz *)

Formula

a(n) = A000123( A000045(n) ) for n>=0.

A155076 Triangle read by rows. The main diagonal is positive. If rowindex >= 2*columnindex then -1 else 0.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, -1, -1, 0, 1, -1, -1, 0, 0, 1, -1, -1, -1, 0, 0, 1, -1, -1, -1, 0, 0, 0, 1, -1, -1, -1, -1, 0, 0, 0, 1, -1, -1, -1, -1, 0, 0, 0, 0, 1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, -1, -1, 0
Offset: 1

Views

Author

Mats Granvik, Gary W. Adamson, Jan 19 2009

Keywords

Comments

Matrix inverse of this triangle is A155077.

Examples

			Table begins:
1,
-1,1,
-1,0,1,
-1,-1,0,1,
-1,-1,0,0,1,
-1,-1,-1,0,0,1,
-1,-1,-1,0,0,0,1,
-1,-1,-1,-1,0,0,0,1,
-1,-1,-1,-1,0,0,0,0,1,

Crossrefs

Cf. A000123.

A162585 G.f.: A(x) = exp( Sum_{n>=1} C(2n,n)*A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 8, 20, 114, 288, 1156, 3256, 23464, 59716, 243212, 699216, 3659988, 10265800, 42353168, 128163440, 1127515970, 2858004752, 11768578868, 34294832344, 180335471424, 513911386232, 2137413847256, 6572758142016, 41948816796852
Offset: 0

Views

Author

Paul D. Hanna, Jul 06 2009

Keywords

Comments

Compare g.f. to the g.f. of the Catalan numbers: exp( Sum_{n>=1} C(2n,n)*x^n/n ), where C(2n,n) form the central binomial coefficients (A000984).

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 146*x^4 + 282*x^5 + 826*x^6 + ...
log(A(x)) = 2*x + 12*x^2/2 + 20*x^3/3 + 280*x^4/4 + 252*x^5/5 + 1848*x^6/6 + ... + C(2n,n)*A006519(n)*x^n/n + ...

Links

Crossrefs

Cf. A000108, A000123, A000984, A006519.

Programs

  • Mathematica
    nmax=50; CoefficientList[Series[Exp[Sum[2^(IntegerExponent[k, 2])*Binomial[2*k, k]*q^k/k, {k,nmax+3}]], {q,0,nmax}], q]  (* G. C. Greubel, Jul 04 2018 *)
  • PARI
    {a(n)=local(L=sum(m=1,n,2^valuation(m,2)*binomial(2*m,m)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}

A162589 G.f.: A(x) = exp( Sum_{n>=1} 2^n*A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 6, 12, 38, 76, 188, 376, 1094, 2188, 5236, 10472, 26076, 52152, 118840, 237680, 612678, 1225356, 2804420, 5608840, 13279604, 26559208, 59074504, 118149008, 277925148, 555850296, 1228260104, 2456520208, 5552652792, 11105305584
Offset: 0

Views

Author

Paul D. Hanna, Jul 07 2009

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 12*x^3 + 38*x^4 + 76*x^5 + 188*x^6 + ...
log(A(x)) = 2*x + 8*x^2/2 + 8*x^3/3 + 64*x^4/4 + 32*x^5/5 + 128*x^6/6 + 128*x^7/7 + ...

Links

Crossrefs

Cf. A162588, A006519, A000123.

Programs

  • Mathematica
    nmax = 150; a[n_]:= SeriesCoefficient[Series[Exp[Sum[2^(k + IntegerExponent[k, 2])*q^k/k, {k, 1, nmax}]], {q,0,nmax}], n]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jul 04 2018 *)
  • PARI
    {a(n)=local(L=sum(m=1,n,2^(m+valuation(m,2))*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}

A180591 G.f.: A(x) = exp( Sum_{n>=1} 2^[A001511(n)^2]*x^n/n ) where A001511(n) is the exponent in the highest power of 2 that divides 2n.

Original entry on oeis.org

1, 2, 10, 18, 178, 338, 1450, 2562, 23234, 43906, 186602, 329298, 2276914, 4224530, 16898506, 29572482, 191488770, 353405058, 1394069578, 2434734098, 14073489714, 25712245330, 97969052778, 170225860226, 938475356354
Offset: 0

Views

Author

Paul D. Hanna, Sep 10 2010

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 18*x^3 + 178*x^4 + 338*x^5 +...
log(A(x)) = 2^1*x + 2^4*x^2/2 + 2^1*x^3/3 + 2^9*x^4/4 + 2^1*x^5/5 + 2^4*x^6/6 + 2^1*x^7/7 + 2^16*x^8/8 +...+ 2^[A001511(n)^2]*x^n/n +...

Crossrefs

Cf. A155200, A001511, A000123.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,2^(valuation(2*m,2)^2)*x^m/m)+x*O(x^n)),n)}

Extensions

Name corrected by Paul D. Hanna, Sep 19 2010

A183034 G.f.: A(x) = exp( Sum_{n>=1} -(-2)^A001511(n)*x^n/n ) where A001511(n) equals the 2-adic valuation of 2n.

Original entry on oeis.org

1, 2, 0, -2, 2, 6, 0, -6, 0, 6, 0, -6, -2, 2, 0, -2, 2, 6, 0, -6, 6, 18, 0, -18, 0, 18, 0, -18, -6, 6, 0, -6, 0, 6, 0, -6, 6, 18, 0, -18, 0, 18, 0, -18, -6, 6, 0, -6, -2, 2, 0, -2, 2, 6, 0, -6, 0, 6, 0, -6, -2, 2, 0, -2, 2, 6, 0, -6, 6, 18, 0, -18, 0, 18, 0, -18, -6, 6, 0, -6, 6, 18, 0, -18, 18, 54, 0
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2010

Keywords

Comments

Compare g.f. to B(x), the g.f. of the number of partitions of 2n into powers of 2 (A000123):
B(x) = exp( Sum_{n>=1} 2^A001511(n)*x^n/n ) = (1-x)^(-1)*Product_{n>=0} 1/(1 - x^(2^n)).

Examples

			G.f.: A(x) = 1 + 2*x - 2*x^3 + 2*x^4 + 6*x^5 - 6*x^7 + 6*x^9 -+...
The logarithm of the g.f. begins:
log(A(x)) = 2*x - 4*x^2/2 + 2*x^3/3 + 8*x^4/4 + 2*x^5/5 - 4*x^6/6 + 2*x^7/7 - 16*x^8/8 + 2*x^9/9 - 4*x^10/10 + 2*x^11/11 + 8*x^12/12 + 2*x^13/13 - 4*x^14/14 + 2*x^15/15 + 32*x^16/16 +...
The g.f. may be expressed by the series:
A(x) = 1 + 2*G(x) + 2*G(x^4) + 2*G(x^16) + 2*G(x^64) + 2*G(x^256) +...
where G(x) is the g.f. of A183035:
G(x) = x*(1-x^2)*Product_{n>=1} (1 + x^(4^n))^3
which begins:
G(x) = x - x^3 + 3*x^5 - 3*x^7 + 3*x^9 - 3*x^11 + x^13 - x^15 + 3*x^17 - 3*x^19 + 9*x^21 - 9*x^23 + 9*x^25 - 9*x^27 + 3*x^29 - 3*x^31 +...

Crossrefs

Cf. A183035, A001511, A000123.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,-(-2)^valuation(2*m,2)*x^m/m)+x*O(x^n)),n)}
  • PARI
    {a(n)=local(L4n=ceil(log(n+1)/log(4)),G=x*(1-x^2)*prod(k=1,L4n,1 + x^(4^k))^3);polcoeff(1+2*sum(k=0,L4n,subst(G,x,x^(4^k)+x*O(x^n))),n)}

Formula

G.f. satisfies: A(x) = A(x^4)*(1+x)^2/(1+x^2).
G.f.: A(x) = 1 + 2*Sum_{n>=0} G(x^(4^n)) where G(x) = x*(1-x^2)*Product_{n>=1} (1 + x^(4^n))^3 is the g.f. of A183035.
a(4n) = a(n); a(4n+2) = 0.

A212775 Number of partitions of 2^(2^n) into powers of 2.

Original entry on oeis.org

2, 4, 36, 692004, 114788185359199234852802340
Offset: 0

Views

Author

Alois P. Heinz, May 26 2012

Keywords

Comments

Lengths (in decimal digits) of the terms a(0), a(1), ... are: 1, 1, 2, 6, 27, 119, 525, 2241, 9330, ... .

Examples

			a(0) = 2 because the number of partitions of 2^2^0 = 2 into powers of 2 is 2: [2], [1,1].
a(1) = 4: [4], [2,2], [2,1,1], [1,1,1,1].

Links

Crossrefs

Cf. A000123, A002577, A172288, A182135.

Programs

  • Maple
    b:= proc(n, j) option remember; local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n b(1, 2^n):
seq(a(n), n=0..6);
  • Mathematica
    b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n<0, 0, j==0, 1, j==1, n+1, n < j, b[n, j] = b[n-1, j] + b[2*n, j-1], True, nn = 1+Floor[n]; r = n-nn; (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)*b[j-h+r, j]*(-1)^h, {h, 0, j-1}]]]; a[n_] := b[1, 2^n]; Table[a[n], {n, 0, 6}] (* Jean-François Alcover, Feb 28 2017, translated from Maple *)

Formula

a(n) = [x^2^(2^n-1)] 1/(1-x) * 1/Product_{j=0..2^n-1} (1-x^(2^j)).

A301702 a(n) = [x^n] Product_{k>=0} 1/(1 - x^(2^k))^n.

Original entry on oeis.org

1, 1, 5, 19, 89, 401, 1877, 8821, 41969, 200899, 967605, 4681491, 22739705, 110816343, 541561333, 2653061819, 13024808161, 64063300481, 315624211781, 1557318893473, 7694243895289, 38060959885345, 188482408625373, 934323819631893, 4635781966972721, 23020536772620401
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2018

Keywords

Comments

Number of binary partitions of n into parts of n kinds.

Links

Crossrefs

Cf. A000123, A008485, A018819, A038712, A171238.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^(2^k))^n, {k, 0, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[(1 + x^(2^k))^(n (k + 1)), {k, 0, n}], {x, 0, n}], {n, 0, 25}]

A304909 Expansion of x * (d/dx) Product_{k>=0} 1/(1 - x^(2^k)).

Original entry on oeis.org

0, 1, 4, 6, 16, 20, 36, 42, 80, 90, 140, 154, 240, 260, 364, 390, 576, 612, 828, 874, 1200, 1260, 1628, 1702, 2256, 2350, 2964, 3078, 3920, 4060, 4980, 5146, 6464, 6666, 8092, 8330, 10224, 10508, 12540, 12870, 15600, 15990, 18900, 19350, 23056, 23580, 27508, 28106, 33216, 33908
Offset: 0

Views

Author

Ilya Gutkovskiy, May 20 2018

Keywords

Comments

Sum of all parts of all partitions of n into powers of 2.
Convolution of the sequences A018819 and A038712.

Links

Crossrefs

Cf. A000079, A000123, A018819, A038712, A066186, A281688, A304908.

Programs

  • Mathematica
    nmax = 49; CoefficientList[Series[x D[Product[1/(1 - x^2^k), {k, 0, Floor[Log[nmax]/Log[2]] + 1}], x], {x, 0, nmax}], x]
nmax = 49; CoefficientList[Series[x D[Product[(1 + x^2^k)^(k + 1), {k, 0, Floor[Log[nmax]/Log[2]] + 1}], x], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Boole[d == 2^IntegerExponent[d, 2]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n a[n], {n, 0, 49}]

Formula

G.f.: x * (d/dx) Product_{k>=0} (1 + x^(2^k))^(k+1).
G.f.: Sum_{i>=0} 2^i*x^(2^i)/(1 - x^(2^i)) * Product_{j>=0} 1/(1 - x^(2^j)).
a(n) = n*A018819(k).
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