A177449 -id:A177449 - cp's OEIS frontend

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

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

Offset: 0

Paul D. Hanna, May 09 2010

nonn

Column 1 of triangle A215241.

			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; ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

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

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

{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)}

{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],", "))}

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

A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.

Original entry on oeis.org

1, 1, 2, 9, 70, 805, 12480, 245847, 5909338, 168310515, 5556486450, 209003251240, 8835266400450, 415094928861530, 21473740362658640, 1213683089969940075, 74446121738526773490, 4927385997649620215895, 350145746700442604768346

Offset: 0

Paul D. Hanna, May 09 2010

nonn

			1+x = 1 + 1*x/(1+x)^2 + 2*x^2/(1+x)^6 + 9*x^3/(1+x)^12 + 70*x^4/(1+x)^20 + 805*x^5/(1+x)^30 +...
1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^4 + 9*x^3*(1-x)^9 + 70*x^4*(1-x)^16 + 805*x^5*(1-x)^25 +...
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 2(n+1) times, starting with a '1' in row 0, as illustrated by:
  1;
  1, 1;
  1, 2,  2,  2,  2;
  1, 3,  5,  7,  9,  9,   9,   9,   9,   9;
  1, 4,  9, 16, 25, 34,  43,  52,  61,  70,  70,  70,  70,  70,  70,  70,  70;
  1, 5, 14, 30, 55, 89, 132, 184, 245, 315, 385, 455, 525, 595, 665, 735, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..356

Cf. A107877, A177447, A177448, A177449, A209440.

a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial(1+ j^2, n-j), j=0..n-1))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 08 2022

{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*(k+1))),n)}

G.f.: Sum_{n>=0} a(n)*x^n*(1-x)^(n^2) = 1/(1-x).
G.f.: Sum_{n>=0} a(n)*x^n*C(-x)^(n^2+2n) = 1/C(-x) where C(x) is the Catalan function of A000108.
a(n) = number of subpartitions of partition consisting of the first n square numbers starting with zero for n>0; e.g., a(4) = subp([0,1,4,9]) = 70. See A115728 for the definition of subpartitions.

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

Paul D. Hanna, May 09 2010

nonn

			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 +...

Cf. A177447, A177449, A177450.

{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)}

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.

cp's OEIS Frontend

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

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A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.

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

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cp's OEIS Frontend

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

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Maple

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PARI

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A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.

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

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