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 1-5 of 5 results.

A177447 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2) = 1+x.

Original entry on oeis.org

1, 1, 1, 3, 18, 172, 2313, 40626, 887326, 23282964, 715540140, 25259729071, 1008721104654, 45008479039824, 2221170817590696, 120209722115431950, 7083266027910364710, 451620678137942740132, 30990400538494184551692, 2277988537997377457967690, 178626191260072536476398000
Offset: 0

Views

Author

Paul D. Hanna, May 09 2010

Keywords

Comments

Column 1 of triangle A215241.

Examples

			1+x = 1 + 1*x/(1+x) + 1*x^2/(1+x)^4 + 3*x^3/(1+x)^9 + 18*x^4/(1+x)^16 + 172*x^5/(1+x)^25 + 2313*x^6/(1+x)^36 +...
Also forms the final terms in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated 2n+1 times, starting with a '1' in row 0, as illustrated by:
  1;
  1, 1,  1;
  1, 2,  3,  3,  3,   3,   3;
  1, 3,  6,  9, 12,  15,  18,  18,  18,  18,  18,  18,  18;
  1, 4, 10, 19, 31,  46,  64,  82, 100, 118, 136, 154, 172,  172,  172,  172,  172,  172,  172,  172,  172;
  1, 5, 15, 34, 65, 111, 175, 257, 357, 475, 611, 765, 937, 1109, 1281, 1453, 1625, 1797, 1969, 2141, 2313, 2313, 2313, 2313, 2313, 2313, 2313, 2313, 2313; ...

Links

Crossrefs

Cf. A215241, A215242, A215243, A133316, A177448, A177449, A177450.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial(1 +j*(j-1), n-j), j=0..n-1))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 10 2022
  • PARI
    {a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(k^2)),n)}
  • PARI
    {A=[1,1];for(i=1,40,A=concat(A,-Vec(sum(n=0,#A-1,A[n+1]*x^n/(1+x+x*O(x^#A))^(n^2)))[#A+1]));for(n=0,#A-1,print1(A[n+1],", "))}

Formula

a(n) = number of subpartitions of the partition [0,0,2,6,12,...,(n-1)^2-(n-1)] for n>0 with a(0)=1. See A115728 for the definition of subpartitions.
Generating functions:
(1) 1 + x = Sum_{n>=0} a(n) * x^n / (1+x)^(n^2).
(2) 1/(1-x) = Sum_{n>=0} a(n) * x^n * (1-x)^(n*(n-1)). - Paul D. Hanna, Apr 04 2022

A209440 G.f.: 1 = Sum_{n>=0} a(n)*x^n * (1-x)^((n+1)^2).

Original entry on oeis.org

1, 1, 4, 30, 340, 5235, 102756, 2464898, 70120020, 2313120225, 86962820000, 3674969314090, 172615622432040, 8928295918586815, 504561763088722500, 30946605756915149850, 2048137516834986743700, 145535818715694311408181, 11054204297079333714850260
Offset: 0

Views

Author

Paul D. Hanna, Apr 07 2012

Keywords

Comments

Compare to a g.f. of the Catalan numbers: 1 = Sum_{n>=0} A000108(n)*x^n*(1-x)^(n+1).

Examples

			G.f.: 1 = 1*(1-x) + 1*x*(1-x)^4 + 4*x^2*(1-x)^9 + 30*x^3*(1-x)^16 + 340*x^4*(1-x)^25 +...

Links

Crossrefs

Cf. A000108, A107878, A177450, A180716.
Column k=2 of A355614.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial((j+1)^2, n-j), j=0..n-1))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 08 2022
  • Mathematica
    a[0] := 1; a[n_] := a[n] = Sum[(-1)^(n + 1 - k)*a[k]*Binomial[(k + 1)^2, n - k], {k, 0, n - 1}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 02 2018 *)
  • PARI
    {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*(1-x+x*O(x^n))^((m+1)^2)), n))}
  • PARI
    {a(n)=if(n==0,1,sum(k=0,n-1,(-1)^(n+1-k)*a(k)*binomial((k+1)^2,n-k)))}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n-1} (-1)^(n+1-k) * a(k) * binomial((k+1)^2,n-k) for n>=1, with a(0)=1.

A177448 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(2*n^2) = 1+x.

Original entry on oeis.org

1, 1, 2, 13, 166, 3324, 92718, 3354712, 150206430, 8050991676, 504049958320, 36172232930282, 2931474921768206, 265078092222575572, 26480336590135734816, 2898139377307388441520, 345055687960080723910286
Offset: 0

Views

Author

Paul D. Hanna, May 09 2010

Keywords

Examples

			1+x = 1 + 1*x/(1+x)^2 + 2*x^2/(1+x)^8 + 13*x^3/(1+x)^18 + 166*x^4/(1+x)^32 + 3324*x^5/(1+x)^50 + 92718*x^6/(1+x)^72 +...

Crossrefs

Cf. A177447, A177449, A177450.

Programs

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

Formula

a(n) = number of subpartitions of the partition [0,1,6,15,28,...,2(n-1)^2-(n-1)] for n>0 with a(0)=1. See A115728 for the definition of subpartitions.

A177449 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(3*n^2) = 1+x.

Original entry on oeis.org

1, 1, 3, 30, 586, 17865, 756285, 41440056, 2805638310, 227131872654, 21459076173105, 2322336372705030, 283667666439112350, 38643426990067599005, 5813534115429573742587, 957883907138024944675200
Offset: 0

Views

Author

Paul D. Hanna, May 09 2010

Keywords

Examples

			1+x = 1 + 1*x/(1+x)^3 + 3*x^2/(1+x)^12 + 30*x^3/(1+x)^27 + 586*x^4/(1+x)^48 + 17865*x^5/(1+x)^75 + 756285*x^6/(1+x)^108 +...

Crossrefs

Cf. A177447, A177448, A177450.

Programs

  • PARI
    {a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(3*k^2)),n)}

Formula

a(n) = number of subpartitions of the partition [0,2,10,24,44,...,3(n-1)^2-(n-1)] for n>0 with a(0)=1. See A115728 for the definition of subpartitions.

A337849 G.f.: 1 = Sum_{n>=0} a(n) * x^n / (1 + x*(1+x)^n)^(n+1).

Original entry on oeis.org

1, 1, 1, 3, 9, 45, 237, 1591, 11795, 99651, 928507, 9474043, 104866399, 1249073623, 15914416345, 215724860511, 3097002496225, 46904398032017, 746831626559889, 12463977258581151, 217445976215654257, 3956098180940284169, 74897945785223884653
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2020

Keywords

Examples

			G.f.: 1 = 1/(1 + x) + x/(1 + x*(1+x))^2 + x^2/(1 + x*(1+x)^2)^3 + 3*x^3/(1 + x*(1+x)^3)^4 + 9*x^4/(1 + x*(1+x)^4)^5 + 45*x^5/(1 + x*(1+x)^5)^6 + 237*x^6/(1 + x*(1+x)^6)^7 + 1591*x^7/(1 + x*(1+x)^7)^8 + 11795*x^8/(1 + x*(1+x)^8)^9 + 99651*x^9/(1 + x*(1+x)^9)^10 + ... + a(n)*x^n/(1 + x*(1+x)^n)^(n+1) + ...

Links

Crossrefs

Cf. A177450.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = -polcoeff( sum(m=0,#A-1, A[m+1] * x^m / (1 + x*(1+x)^m +O(x^#A) )^(m+1) ), #A-1) );A[n+1]}
\\ Print initial terms:
for(n=0,30,print1(a(n),", "))
\\ Verify sum equals unity:
sum(n=0,40, a(n)*x^n / (1 + x*(1+x)^n +O(x^41) )^(n+1) )

Formula

G.f.: 1 = Sum_{n>=0} a(n) * x^n / (1 + x*(1+x)^n)^(n+1).
G.f.: 1 = Sum_{n>=0} a(n) * x^n * (1-x)^(n^2+n+1) / ((1-x)^(n+1) + x)^(n+1).
Showing 1-5 of 5 results.

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