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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A375448 Expansion of the g.f. A(x) with the property that the sum of the first n coefficients in A(x/n)^n equals n^2 for n >= 1.

Original entry on oeis.org

1, 3, 6, 33, 357, 5283, 96534, 2067312, 50345955, 1367512761, 40875976152, 1331343423234, 46892513148468, 1775323414999818, 71885746640828286, 3100014000785085216, 141857882269044077865, 6866221878372182554395, 350521791594556907681202, 18824690900373744731703396
Offset: 0

Views

Author

Paul D. Hanna, Sep 12 2024

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 6*x^2 + 33*x^3 + 357*x^4 + 5283*x^5 + 96534*x^6 + 2067312*x^7 + 50345955*x^8 + 1367512761*x^9 + 40875976152*x^10 + ...
The defining property of g.f. A(x) is described below.
The table of coefficients in A(x)^n begins:
  n=1: [1,  3,   6,   33,   357,   5283,   96534, ...];
  n=2: [1,  6,  21,  102,   948,  13104,  230139, ...];
  n=3: [1,  9,  45,  234,  1935,  24678,  414234, ...];
  n=4: [1, 12,  78,  456,  3561,  41868,  667746, ...];
  n=5: [1, 15, 120,  795,  6150,  67428, 1017540, ...];
  n=6: [1, 18, 171, 1278, 10107, 105246, 1501578, ...];
  n=7: [1, 21, 231, 1932, 15918, 160587, 2172807, ...];
  ...
in which the sum of the first n coefficients in A(x/n)^n equals n^2, as illustrated by
  1 = 1;
  4 = 1 + 6/2;
  9 = 1 + 9/3 + 45/3^2;
  16 = 1 + 12/4 + 78/4^2 + 456/4^3;
  25 = 1 + 15/5 + 120/5^2 + 795/5^3 + 6150/5^4;
  36 = 1 + 18/6 + 171/6^2 + 1278/6^3 + 10107/6^4 + 105246/6^5;
  49 = 1 + 21/7 + 231/7^2 + 1932/7^3 + 15918/7^4 + 160587/7^5 + 2172807/7^6;
  ...
RELATED SERIES.
Let B(x) be the series reversion of x/A(x), B(x/A(x)) = x, then
B(x) = x + 3*x^2 + 15*x^3 + 114*x^4 + 1230*x^5 + 17541*x^6 + 310401*x^7 + 6502368*x^8 + ... + A375449(n)*x^n + ...
Further, let C(x) = x*B'(x)/(1 - B(x)) = x + 7*x^2 + 55*x^3 + 547*x^4 + 7081*x^5 + 116821*x^6 + 2351497*x^7 + 55390315*x^8 + ...
then the coefficient of x^n in C(x) equals the sum of the initial n terms of A(x)^n for n >= 1; 1 = 1, 7 = 1 + 6, 55 = 1 + 9 + 45, 547 = 1 + 12 +  78 + 456, 7081 = 1 + 15 + 120 + 795 + 6150, etc.

Links

Crossrefs

Cf. A375449, A375457.

Programs

  • PARI
    {a(n) = my(A=[1], m, V); for(i=0, n, A = concat(A, 0); m=#A; V=Vec( subst(Ser(A)^m, x, x/m) );
A[m] = (m^2 - sum(k=1, #V, V[k]) )*m^(m-2) ); H=A; A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) satisfies [x^n] x*B'(x/n) / (1 - n*B(x/n)) = n^2 for n >= 1, where B(x/A(x)) = x and B(x) is the g.f. of A375449.
a(n) ~ c * n^(n+1), where c = 0.5698891168602..., conjecture: c = (exp(1)-1)/exp(3*exp(-1)). - Vaclav Kotesovec, Sep 13 2024

A375452 Expansion of g.f. A(x) satisfying [x^n] x*A'(x) / (1 - n*A(x)) = n^n for n >= 1.

Original entry on oeis.org

1, 1, 3, 18, 170, 2181, 34909, 663152, 14493060, 356894730, 9757244200, 292895688618, 9570802060144, 338073560791701, 12835113747145545, 521191511664816256, 22541443155867228040, 1034573007453548334066, 50226021692317694266696, 2571751515973758412301940
Offset: 1

Views

Author

Paul D. Hanna, Sep 09 2024

Keywords

Comments

Conjecture: for n > 1, a(n) is odd iff n is of the form 2^k-2 or 2^k-1 for some k > 1.
Compare to [x^n] x*G'(x) / (1 - n*G(x)) = ((n+1)^n - 1)/n for n >= 1 when G(x) = x/(1-x).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 18*x^4 + 170*x^5 + 2181*x^6 + 34909*x^7 + 663152*x^8 + 14493060*x^9 + 356894730*x^10 + 9757244200*x^11 + 292895688618*x^12 + ...
where [x^n] x*A'(x) / (1 - n*A(x)) = n^n for n >= 1.
The table of coefficients of x^k in x*A'(x)/(1 - n*A(x)) begins
  n=1: [1, 3, 13,  91,  981, 14421, 262963, ...];
  n=2: [1, 4, 19, 124, 1196, 16324, 286553, ...];
  n=3: [1, 5, 27, 177, 1561, 19323, 319159, ...];
  n=4: [1, 6, 37, 256, 2166, 24330, 368887, ...];
  n=5: [1, 7, 49, 367, 3125, 32761, 450563, ...];
  n=6: [1, 8, 63, 516, 4576, 46656, 589093, ...];
  n=7: [1, 9, 79, 709, 6681, 68799, 823543, ...];
  ...
in which the main diagonal equals n^n for n >= 1.
RELATED SERIES.
x*A'(x)/(1 - A(x)) = x + 3*x^2 + 13*x^3 + 91*x^4 + 981*x^5 + 14421*x^6 + 262963*x^7 + 5630843*x^8 + ...
Let B(x) satisfy A(x/B(x)) = x, then B(x) begins
B(x) = 1 + x + 2*x^2 + 11*x^3 + 105*x^4 + 1375*x^5 + 22390*x^6 + 430954*x^7 + 9512029*x^8 + ... + A375457(n)*x^n + ...
Below we demonstrate some properties of this related series.
The table of coefficients in B(x)^n begins:
  n=1: [1, 1,  2,  11,  105,  1375, ...];
  n=2: [1, 2,  5,  26,  236,  3004, ...];
  n=3: [1, 3,  9,  46,  399,  4932, ...];
  n=4: [1, 4, 14,  72,  601,  7212, ...];
  n=5: [1, 5, 20, 105,  850,  9906, ...];
  ...
in which the sum of the first n coefficients in B(x)^n equals the coefficients in x*A'(x)/(1 - A(x)) like so: 1 = 1, 3 = 1 + 2, 13 = 1 + 3 + 9, 91 = 1 + 4 + 14 + 72, 981 = 1 + 5 + 20 + 105 + 850, etc.
Also, the sum of the first n coefficients in B(x/n)^n equals n, as illustrated by
 1 = 1;
 2 = 1 + 2/2;
 3 = 1 + 3/3 + 9/3^2;
 4 = 1 + 4/4 + 14/4^2 + 72/4^3;
 5 = 1 + 5/5 + 20/5^2 + 105/5^3 + 850/5^4;
 ...

Links

Crossrefs

Cf. A375449, A375457.

Programs

  • PARI
    {a(n) = my(A=[0,1],Ax=x,m); for(i=1,n, A = concat(A,0); Ax = Ser(A); m=#A-1; A[#A] = (m^m - polcoef( x*Ax' / (1 - m*Ax), m) )/m ); A[n+1]}
for(n=1,20,print1(a(n),", "))

Formula

a(n) ~ (1 - exp(-1)) * n^(n-1). - Vaclav Kotesovec, Sep 10 2024
Showing 1-2 of 2 results.

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