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 1-2 of 2 results.

A375449 Expansion of g.f. A(x) satisfying [x^n] x*A'(x) / (1 - n*A(x)) = n^(n+1) for n >= 1.

Original entry on oeis.org

1, 3, 15, 114, 1230, 17541, 310401, 6502368, 156119778, 4202894190, 124940121966, 4056054468282, 142598505603138, 5393588598473229, 218309634723396555, 9413799845705265408, 430826174378500622916, 20856810739070097320814, 1064963403446152512517140, 57204475911565427208990300
Offset: 1

Views

Author

Paul D. Hanna, Sep 12 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 + 3*x^2 + 15*x^3 + 114*x^4 + 1230*x^5 + 17541*x^6 + 310401*x^7 + 6502368*x^8 + 156119778*x^9 + 4202894190*x^10 + ...
The table of coefficients of x^k in x*A'(x)/(1 - n*A(x)) begins
  n=1: [1,  7,  55,  547,  7081, 116821, 2351497, ...];
  n=2: [1,  8,  67,  668,  8356, 132260, 2577541, ...];
  n=3: [1,  9,  81,  825, 10101, 153387, 2874537, ...];
  n=4: [1, 10,  97, 1024, 12466, 182698, 3278227, ...];
  n=5: [1, 11, 115, 1271, 15625, 223481, 3840817, ...];
  n=6: [1, 12, 135, 1572, 19776, 279936, 4636017, ...];
  n=7: [1, 13, 157, 1933, 25141, 357295, 5764801, ...];
  ...
in which the main diagonal equals n^(n+1) for n >= 1.
RELATED SERIES.
x*A'(x)/(1 - A(x)) = x + 7*x^2 + 55*x^3 + 547*x^4 + 7081*x^5 + 116821*x^6 + 2351497*x^7 + 55390315*x^8 + ...
Let B(x) satisfy A(x/B(x)) = x, then B(x) begins
B(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 + ... + A375448(n)*x^n + ...
Below we demonstrate some properties of this related series.
The table of coefficients in B(x)^n begins:
  n=1: [1,  3,   6,  33,  357,  5283, ...];
  n=2: [1,  6,  21, 102,  948, 13104, ...];
  n=3: [1,  9,  45, 234, 1935, 24678, ...];
  n=4: [1, 12,  78, 456, 3561, 41868, ...];
  n=5: [1, 15, 120, 795, 6150, 67428, ...];
  ...
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, 7 = 1 + 6, 55 = 1 + 9 + 45, 547 = 1 + 12 +  78 + 456, 7081 = 1 + 15 + 120 + 795 + 6150, etc.
Also, the sum of the first n coefficients in B(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;
 ...

Links

Crossrefs

Cf. A375448, A375452.

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+1) - 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. - Vaclav Kotesovec, Sep 13 2024

A375457 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 for n >= 1.

Original entry on oeis.org

1, 1, 2, 11, 105, 1375, 22390, 430954, 9512029, 235992263, 6488607220, 195627162152, 6414053158664, 227170447034030, 8643069830739980, 351580969750713450, 15228097928340597681, 699791999466718937425, 34010355409897760336176, 1743142054929355666550574, 93975675621720312817066020
Offset: 0

Views

Author

Paul D. Hanna, Sep 08 2024

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 105*x^4 + 1375*x^5 + 22390*x^6 + 430954*x^7 + 9512029*x^8 + 235992263*x^9 + 6488607220*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, 1,  2,  11,  105,  1375,  22390, ...];
  n=2: [1, 2,  5,  26,  236,  3004,  48071, ...];
  n=3: [1, 3,  9,  46,  399,  4932,  77498, ...];
  n=4: [1, 4, 14,  72,  601,  7212, 111194, ...];
  n=5: [1, 5, 20, 105,  850,  9906, 149760, ...];
  n=6: [1, 6, 27, 146, 1155, 13086, 193886, ...];
  n=7: [1, 7, 35, 196, 1526, 16835, 244363, ...];
  ...
in which the sum of the first n coefficients in A(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;
  6 = 1 + 6/6 + 27/6^2 + 146/6^3 + 1155/6^4 + 13086/6^5;
  7 = 1 + 7/7 + 35/7^2 + 196/7^3 + 1526/7^4 + 16835/7^5 + 244363/7^6;
  ...
RELATED SERIES.
Let B(x) be the series reversion of x/A(x), B(x/A(x)) = x, then
B(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 + ... + A375452(n)*x^n + ...
Further, let C(x) = x*B'(x)/(1 - B(x)) = x + 3*x^2 + 13*x^3 + 91*x^4 + 981*x^5 + 14421*x^6 + 262963*x^7 + 5630843*x^8 + 137203969*x^9 + ...
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, 3 = 1 + 2, 13 = 1 + 3 + 9, 91 = 1 + 4 + 14 + 72, 981 = 1 + 5 + 20 + 105 + 850, etc.
The logarithmic derivative of g.f. A(x) begins
A(x)'/A(x) = 1 + 3*x + 28*x^2 + 375*x^3 + 6306*x^4 + 125286*x^5 + 2845200*x^6 + 72355095*x^7 + 2031897160*x^8 + 62371350558*x^9 + 2076430998588*x^10 + ...
Notice that the coefficient of x^n in A(x)'/A(x) appears to be divisible by (n+2) for n > 0.

Links

Crossrefs

Cf. A375452, A375448, A088358.

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 - sum(k=1,#V,V[k]) )*m^(m-2) ); 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 for n >= 1, where B(x/A(x)) = x and B(x) is the g.f. of A375452.
a(n) ~ c * n^n, where c = 1.189395759976..., conjecture: c = (exp(1)-1)/exp(exp(-1)). - Vaclav Kotesovec, Sep 13 2024
Showing 1-2 of 2 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.