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 1-8 of 8 results.

A300647 Number of same-trees of weight n in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 10, 2, 2, 2, 2, 2, 42, 1, 2, 10, 2, 2, 138, 2, 2, 2, 34, 2, 1514, 2, 2, 42, 2, 1, 2058, 2, 162, 10, 2, 2, 8202, 2, 2, 138, 2, 2, 207370, 2, 2, 2, 130, 34, 131082, 2, 2, 1514, 2082, 2, 524298, 2, 2, 42, 2, 2, 14725738, 1, 8226, 2058, 2
Offset: 1

Views

Author

Gus Wiseman, Mar 10 2018

Keywords

Comments

A same-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more same-trees whose weights are all equal and sum to n.

Examples

			The a(9) = 10 odd same-trees:
9,
(333),
(33(111)), (3(111)3), ((111)33)
(3(111)(111)), ((111)3(111)), ((111)(111)3),
((111)(111)(111)), (111111111).

Links

Crossrefs

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300648, A300649, A300650.

Programs

  • Mathematica
    a[n_]:=1+Sum[a[n/d]^d,{d,Select[Rest[Divisors[n]],OddQ]}];
Array[a,80]
  • PARI
    a(n) = if (n==1, 1, 1+sumdiv(n, d, if ((d > 1) && (d % 2), a(n/d)^d))); \\ Michel Marcus, Mar 10 2018

Formula

a(n) = 1 + Sum_d a(n/d)^d where the sum is over odd divisors of n greater than 1.

A300648 Number of orderless same-trees of weight n in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 12, 1, 2, 6, 2, 2, 14, 2, 2, 2, 8, 2, 68, 2, 2, 12, 2, 1, 18, 2, 16, 6, 2, 2, 20, 2, 2, 14, 2, 2, 644, 2, 2, 2, 10, 8, 24, 2, 2, 68, 20, 2, 26, 2, 2, 12, 2, 2, 1386, 1, 22, 18, 2, 2, 30, 16, 2, 6, 2, 2, 4532, 2, 22, 20
Offset: 1

Views

Author

Gus Wiseman, Mar 10 2018

Keywords

Comments

An orderless same-tree of weight n > 0 is either a single node of weight n, or a finite multiset of two or more orderless same-trees whose weights are all equal and sum to n.

Examples

			The a(9) = 6 odd orderless same-trees: 9, (333), (33(111)), (3(111)(111)), ((111)(111)(111)), (111111111).

Links

Crossrefs

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300647, A300649, A300650.

Programs

  • Mathematica
    a[n_]:=1+Sum[Binomial[a[n/d]+d-1,d],{d,Select[Rest[Divisors[n]],OddQ]}];
Array[a,80]
  • PARI
    a(n) = if (n==1, 1, 1 + sumdiv(n, d, if ((d > 1) && (d % 2), binomial(a(n/d) + d - 1, d)))); \\ Michel Marcus, Mar 10 2018

Formula

a(n) = 1 + Sum_d binomial(a(n/d) + d - 1, d) where the sum is over odd divisors of n greater than 1.

A300575 Coefficient of x^n in (1+x)(1-x^3)(1+x^5)(1-x^7)(1+x^9)...

Original entry on oeis.org

1, 1, 0, -1, -1, 1, 1, -1, -2, 0, 2, 0, -3, -1, 3, 2, -3, -3, 3, 4, -3, -6, 2, 7, -1, -8, 0, 10, 2, -11, -4, 12, 7, -13, -10, 13, 13, -13, -17, 13, 22, -11, -26, 9, 31, -6, -36, 2, 41, 3, -46, -9, 51, 17, -55, -26, 59, 36, -62, -48, 63, 61, -64, -75, 64, 92, -60, -109, 55, 127, -48, -147, 37, 167
Offset: 0

Views

Author

Gus Wiseman, Mar 09 2018

Keywords

Links

Crossrefs

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A220418, A290261, A292043, A292137, A300574.

Programs

  • Mathematica
    CoefficientList[Series[QPochhammer[-x,-x^2],{x,0,100}],x]

Formula

O.g.f.: Product_{n >= 0} (1 + (-1)^n x^(2n+1)).
a(n) = Sum (-1)^k where the sum is over all strict integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.

A300649 Number of same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 10, 1, 1, 3, 3, 1, 3, 1, 1, 62, 1, 2, 3, 1, 3, 3, 1, 1, 158, 3, 1, 3, 1, 1, 254, 3, 1, 1514, 1, 3, 3, 1, 3, 3, 3, 1, 2078, 1, 1, 2461, 1, 1, 3, 1, 3, 8222, 3, 2, 3, 34, 1, 3, 1, 3, 390782, 1, 1, 3, 3, 3, 2198, 1, 1
Offset: 0

Views

Author

Gus Wiseman, Mar 10 2018

Keywords

Comments

A same-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more same-trees whose weights are all equal and sum to n.

Examples

			The a(13) = 10 odd same-trees with all leaves greater than 1:
27,
(999),
(99(333)), (9(333)9), ((333)99),
(9(333)(333)), ((333)9(333)), ((333)(333)9),
((333)(333)(333)), (333333333).

Crossrefs

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300647, A300648, A300650.

Programs

  • Mathematica
    a[n_]:=If[n===1,1,Sum[a[n/d]^d,{d,Select[Rest[Divisors[n]],OddQ]}]];
Table[a[n],{n,1,100,2}]
  • PARI
    f(n) = if (n==1, 1, sumdiv(n, d, if ((d > 1) && (d % 2), f(n/d)^d)));
a(n) = f(2*n+1); \\ Michel Marcus, Mar 10 2018

Formula

a(1) = 1; a(n > 1) = Sum_d a(n/d)^d where the sum is over odd divisors of n greater than 1.

A300650 Number of orderless same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 6, 1, 1, 3, 3, 1, 3, 1, 1, 19, 1, 2, 3, 1, 3, 3, 1, 1, 21, 3, 1, 3, 1, 1, 28, 3, 1, 68, 1, 3, 3, 1, 3, 3, 3, 1, 25, 1, 1, 71, 1, 1, 3, 1, 3, 27, 3, 2, 3, 8, 1, 3, 1, 3, 1656, 1, 1, 3, 3, 3, 43, 1, 1, 31, 3, 1, 3, 3, 1
Offset: 0

Views

Author

Gus Wiseman, Mar 10 2018

Keywords

Comments

An orderless same-tree of weight n > 0 is either a single node of weight n, or a finite multiset of two or more orderless same-trees whose weights are all equal and sum to n.

Examples

			The a(13) = 6 orderless same-trees: 27, (999), (99(333)), (9(333)(333)), ((333)(333)(333)), (333333333).

Crossrefs

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300647, A300648, A300649.

Programs

  • Mathematica
    a[n_]:=If[n===1,1,Sum[Binomial[a[n/d]+d-1,d],{d,Select[Rest[Divisors[n]],OddQ]}]];
Table[a[n],{n,1,100,2}]
  • PARI
    f(n) = if (n==1, 1, sumdiv(n, d, if ((d > 1) && (d % 2), binomial(f(n/d)+d-1, d))));
a(n) = f(2*n+1); \\ Michel Marcus, Mar 10 2018

Formula

a(1) = 1; a(n > 1) = Sum_d binomial(a(n/d) + d - 1, d) where the sum is over odd divisors of n greater than 1.

A300652 Number of enriched p-trees of weight 2n + 1 in which all outdegrees and all leaves are odd.

Original entry on oeis.org

1, 2, 4, 12, 40, 136, 496, 1952, 7488, 30368, 123456, 512384, 2129664, 9068672, 38391552, 165642752, 713405952, 3109135872, 13528865792, 59591322624, 261549260800, 1159547047936, 5131968999424, 22883893137408, 101851069587456, 456703499042816, 2042949493276672
Offset: 0

Views

Author

Gus Wiseman, Mar 10 2018

Keywords

Comments

An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of at least two enriched p-trees whose weights are weakly decreasing and sum to n.

Examples

			The a(3) = 12 trees:
7,
(511), (331),
((111)31), (3(111)1), ((311)11), (31111),
((111)(111)1), (((111)11)11), ((11111)11), ((111)1111), (1111111).

Links

Crossrefs

Cf. A000009, A000041, A063834, A196545, A273873, A281145, A289501, A298118, A300352, A300353, A300354, A300436, A300439, A300442, A300443, A300574, A300797.

Programs

  • Mathematica
    r[n_]:=r[n]=If[OddQ[n],1,0]+Sum[Times@@r/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Table[r[n],{n,1,40,2}]
  • PARI
    seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))) - 1/prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ Andrew Howroyd, Aug 26 2018

Formula

a(n) = (1 - (-1)^n)/2 + Sum_y Product_{i in y} a(i) where the sum is over all non-singleton integer partitions of n with an odd number of parts.

A193863 Expansion of Product_{n>=0} (1 + q*(-q^2)^n) / (1 - q*(-q^2)^n).

Original entry on oeis.org

1, 2, 2, 0, -2, 0, 4, 4, -2, -6, 0, 8, 4, -8, -8, 8, 14, -4, -18, 0, 24, 8, -28, -20, 28, 34, -24, -48, 16, 64, 0, -76, -18, 88, 44, -96, -78, 96, 116, -88, -160, 68, 208, -32, -252, -16, 296, 84, -332, -170, 354, 272, -360, -392, 344, 528, -296, -672, 216, 824, -96, -976, -72, 1116, 286, -1240, -552, 1336, 876, -1384
Offset: 0

Views

Author

Joerg Arndt, Aug 07 2011

Keywords

Comments

Expansion of E(-q^2, +q) for E(q,x) = Product_{n>=0} ( 1 + x*q^n ) / ( 1 - x*q^n ).
Replacing q by -q in the g.f. gives the inverse of the g.f., whose expansion is obtained by negating every second term.

Examples

			1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 + 4*x^7 - 2*x^8 - 6*x^9 + 8*x^11 + ...

Links

Crossrefs

Cf. A015128 E(+q,+q), A002448 E(+q,-q), A000122 E(-q,+q), A004402 E(-q,-q), A080054 E(+q^2,+q), A108494 E(+q^2,-q), A300574, A300575.

Programs

  • PARI
    N=66; q='q+O('q^N); /* that many terms */
gf = prod(n=0, N, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) );
Vec(gf) /* show terms */
/* Alternative computation of the g.f. using a product form */
V=[0,-2, 1, 2, 0, -2, -1, 2]; /* note vectors are one-based */
gf=prod(n=0, N, (1-q^n)^(V[n%8+1]) );
  • PARI
    {a(n) = local(A); if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n) )^[ 0, -2, 1, 2, 0, -2, -1, 2][k%8 + 1]), n))} /* Michael Somos, Feb 26 2012 */

Formula

Euler transform of period 8 sequence [ 2, -1, -2, 0, 2, 1, -2, 0, ...]. - Michael Somos, Feb 26 2012
G.f.: prod(n>=0, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) ).
G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k )/prod(k=1..n, 1-(-q^2)^k ) * q^n ).
G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k)/( prod(k=1..n, 1-(-q^2)^k) * prod(k=0..n-1, 1-q*(-q^2)^k ) ) * q^n * (-q^2)^(n*(n-1)/2) ).
Convolution of A300574 and A300575. - Seiichi Manyama, Nov 22 2019

A309240 Expansion of 1/((1 - x)*(1 - x^2)*(1 + x^3)*(1 + x^4)*(1 - x^5)*(1 - x^6)*(1 + x^7)*(1 + x^8)*...).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 3, 4, 2, 4, 4, 7, 5, 7, 6, 11, 9, 13, 10, 17, 14, 20, 15, 25, 22, 32, 24, 36, 31, 48, 38, 55, 45, 68, 55, 79, 65, 97, 79, 112, 91, 136, 113, 159, 128, 186, 156, 221, 179, 256, 213, 301, 245, 347, 290, 409, 334, 466, 388, 547, 451, 624, 517, 724, 600, 828, 687, 955, 793, 1088
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 17 2019

Keywords

Links

Crossrefs

Cf. A000700, A035451, A147599, A300574.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[1/(1 + (-1)^(k (k + 1)/2) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k - 2))/((1 + x^(4 k - 1)) (1 - x^(4 k - 3))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[2/(QPochhammer[-1, -x^2] QPochhammer[x, -x^2]), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 + (-1)^(k*(k+1)/2) * x^k).
G.f.: Product_{k>=1} (1 + x^(4*k-2)) / ((1 + x^(4*k-1)) * (1 - x^(4*k-3))).
a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n/6)) / (4 * 6^(1/8) * Pi^(3/4) * n^(5/8)). - Vaclav Kotesovec, Jul 17 2019
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.