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 1-3 of 3 results.

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

A233436 a(n) = Sum_{k=0..n-1} [x^k] A(x)^(n-1) for n>=1 with a(0)=1, where g.f. A(x) = Sum_{n>=0} a(n)*x^n.

Original entry on oeis.org

1, 1, 2, 8, 50, 424, 4472, 55760, 797022, 12801296, 227829866, 4446822688, 94422531876, 2166975912496, 53457972027254, 1410960809766320, 39680975219789210, 1184783226216138592, 37434788449030871076, 1248022160663960432264, 43785432805297352937954, 1612690422384099635004264
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 424*x^5 + 4472*x^6 + 55760*x^7 +...
ILLUSTRATION OF INITIAL TERMS.
If we form an array of coefficients of x^k in A(x)^n, n>=0, like so:
A^0 = [1],0,  0,   0,    0,     0,      0,       0,        0, ...;
A^1 = [1, 1], 2,   8,   50,   424,   4472,   55760,   797022, ...;
A^2 = [1, 2,  5], 20,  120,   980,  10056,  122960,  1732736, ...;
A^3 = [1, 3,  9,  37], 216,  1704,  17006,  203760,  2829030, ...;
A^4 = [1, 4, 14,  60,  345], 2640,  25632,  300744,  4111472, ...;
A^5 = [1, 5, 20,  90,  515,  3841], 36310,  417000,  5609960, ...;
A^6 = [1, 6, 27, 128,  735,  5370,  49493], 556212,  7359480, ...;
A^7 = [1, 7, 35, 175, 1015,  7301,  65723,  722765], 9400986, ...;
A^8 = [1, 8, 44, 232, 1366,  9720,  85644,  921864, 11782417], ...; ...
then a(n) equals the sum of the coefficients of x^k, k=0..n-1, in A(x)^(n-1) (shown above in brackets) for n>=1:
a(1) = 1 = 1;
a(2) = 1 +  1 = 2;
a(3) = 1 +  2 +  5 = 8;
a(4) = 1 +  3 +  9 +  37 = 50;
a(5) = 1 +  4 + 14 +  60 +  345 = 424;
a(6) = 1 +  5 + 20 +  90 +  515 + 3841 = 4472;
a(7) = 1 +  6 + 27 + 128 +  735 + 5370 + 49493 = 55760;
a(8) = 1 +  7 + 35 + 175 + 1015 + 7301 + 65723 + 722765 = 797022; ...
Also, from a diagonal in the above table we can obtain the coefficients:
[1/1, 2/2, 9/3, 60/4, 515/5, 5370/6, 65723/7, 921864/8, ...]
to form the power series
G(x) = 1 + x + 3*x^2 + 15*x^3 + 103*x^4 + 895*x^5 + 9389*x^6 + 115233*x^7 +...
that satisfies: A(x) = G(x/A(x)) = 1 + x*(G(x) + x*G'(x))/(G(x) - x*G(x)^2).

Links

Crossrefs

Cf. A088713, A088358.

Programs

  • PARI
    {a(n)=local(A=1+x);if(n==0,1,for(i=1,n,
A=1+sum(k=1,n-1,sum(j=0,k-1,polcoeff(A^(k-1)+x*O(x^j),j))*x^k)+x*O(x^n));
sum(j=0,n-1,polcoeff(A^(n-1)+x*O(x^j),j)))}
for(n=0,20,print1(a(n),", "))

Formula

Given g.f. A(x), let G(x) = A(x*G(x)), then A(x) = G(x/A(x)) = 1 + x*(G(x) + x*G'(x)) / (G(x) - x*G(x)^2).
a(n)/a(n-1) ~ n/LambertW(1). - Vaclav Kotesovec, Sep 14 2024

A206801 a(n) equals the sum of the coefficients of x^(n-1) in A(x)^k as k=1..n, where A(x) = Sum_{n>=0} a(n)*x^n.

Original entry on oeis.org

1, 1, 3, 22, 285, 5476, 143080, 4840256, 205348521, 10675415800, 667749546288, 49499829165414, 4292944063388838, 430749877491962328, 49520529524763502478, 6467220964216618610444, 952209872438113961762005, 157000879970348439087083080
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 22*x^3 + 285*x^4 + 5476*x^5 + 143080*x^6 +...
The coefficients in the powers of the g.f. A(x) begin:
A^1: [1, 1,  3,  22,  285,  5476,  143080,  4840256, ...];
A^2: [1, 2,  7,  50,  623, 11654,  299306, 10012068, ...];
A^3: [1, 3, 12,  85, 1023, 18627,  469956, 15539040, ...];
A^4: [1, 4, 18, 128, 1495, 26500,  656450, 21446816, ...];
A^5: [1, 5, 25, 180, 2050, 35391,  860365, 27763280, ...];
A^6: [1, 6, 33, 242, 2700, 45432, 1083451, 34518774, ...];
A^7: [1, 7, 42, 315, 3458, 56770, 1327648, 41746335, ...];
A^8: [1, 8, 52, 400, 4338, 69568, 1595104, 49481952, ...]; ...
where a(n) = sum of the first n coefficients in column n-1 for n>=1:
a(1) = 1,
a(2) = 1 + 2 = 3;
a(3) = 3 + 7 + 12 = 22;
a(4) = 22 + 50 + 85 + 128 = 285;
a(5) = 285 + 623 + 1023 + 1495 + 2050 = 5476; ...

Crossrefs

Cf. A088358.

Programs

  • PARI
    {a(n)=local(A=[1]);for(i=1,n,A=concat(A,sum(m=1,#A,Vec(Ser(A)^m)[#A])));A[n+1]}
for(n=0,21,print1(a(n),", "))

Formula

G.f. A(x) satisfies: a(n) = Sum_{k=1..n} [x^(n-1)] A(x)^k for n>=1, with a(0)=1.
Showing 1-3 of 3 results.

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