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.

A180747 G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (n+1)^(n-1)*x^n.

Original entry on oeis.org

1, 1, 2, 9, 68, 715, 9527, 152789, 2856984, 60962112, 1461364051, 38885737209, 1137587558289, 36299474019445, 1254966476969859, 46739853740801293, 1865947811034153032, 79499993864122690971, 3600874278814894340648
Offset: 0

Views

Author

Paul D. Hanna, Jan 22 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 68*x^4 + 715*x^5 +...
G.f. satisfies A(x) = G(x/A(x)) where G(x) begins:
G(x) = 1 + x + 3*x^2 + 4^2*x^3 + 5^3*x^4 + 6^4*x^5 +...
so that:
A(x) = 1 + x/A(x) + 3*x^2/A(x)^2 + 4^2*x^3/A(x)^3 + 5^3*x^4/A(x)^4 +...
The coefficients in A(x)^n for n=1..8 begin:
A^1: [(1), 1, 2, 9, 68, 715, 9527, 152789, 2856984, ...];
A^2: [1,(2), 5, 22, 158, 1602, 20837, 328716, ...];
A^3: [1, 3,(9), 40, 276, 2700, 34250, 531093, ...];
A^4: [1, 4, 14,(64), 429, 4056, 50146, 763752, ...];
A^5: [1, 5, 20, 95,(625), 5726, 68975, 1031130,  ...];
A^6: [1, 6, 27, 134, 873,(7776), 91268, 1338366, ...];
A^7: [1, 7, 35, 182, 1183, 10283,(117649), 1691411, ...];
A^8: [1, 8, 44, 240, 1566, 13336, 148848,(2097152), ...]; ...
where the coefficient of x^n in A(x)^(n+1) equals (n+1)^n.

Links

Crossrefs

Cf. A182957, A180749.

Programs

  • PARI
    {a(n)=polcoeff(x/serreverse(x*sum(m=0,n+1,(m+1)^(m-1)*x^m)+x^2*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)^n for n>=0.
G.f. satisfies: A(x) = G(x/A(x)) where A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)^(n-1)*x^n.

A182957 G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} n^n*x^n.

Original entry on oeis.org

1, 1, 3, 17, 151, 1824, 27541, 494997, 10273039, 241217147, 6314907390, 182283959604, 5750796304553, 196865960240416, 7268410972604665, 287920792767378837, 12181570018235995359, 548274960053921957856
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2010

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1824*x^5 +...
G.f. satisfies A(x) = G(x/A(x)) where A(x*G(x)) = G(x) begins:
G(x) = 1 + x + 2^2*x^2 + 3^3*x^3 + 4^4*x^4 + 5^5*x^5 + 6^6*x^6 +...
so that:
A(x) = 1 + x/A(x) + 2^2*x^2/A(x)^2 + 3^3*x^3/A(x)^3 + 4^4*x^4/A(x)^4 +...
The coefficients in A(x)^n for n=1..8 begin:
A^1: [(1), 1, 3, 17, 151, 1824, 27541, 494997, ...];
A^2: [1,(2), 7, 40, 345, 4052, 59925, 1061154, ...];
A^3: [1, 3,(12), 70, 591, 6762, 97938, 1707987, ...];
A^4: [1, 4, 18,(108), 899, 10044, 142488, 2446336, ...];
A^5: [1, 5, 25, 155,(1280), 14001, 194620, 3288540, ...];
A^6: [1, 6, 33, 212, 1746,(18750), 255532, 4248630, ...];
A^7: [1, 7, 42, 280, 2310, 24423,(326592), 5342541, ...];
A^8: [1, 8, 52, 360, 2986, 31168, 409356, (6588344), ...]; ...
where the coefficient of x^n in A(x)^(n+1)/(n+1) equals n^n.

Crossrefs

Cf. A180747, A180749.

Programs

  • PARI
    {a(n)=polcoeff(x/serreverse(sum(m=1,n+1,(m-1)^(m-1)*x^m)+x^2*O(x^n)),n)}

Formula

G.f. satisfies: A(x) = G(x/A(x)) where A(x*G(x)) = G(x) = Sum_{n>=0} n^n*x^n.
G.f. satisfies: [x^n] A(x)^(n+1)/(n+1) = n^n.

A242749 G.f. satisfies: A(x) = G(x/A(x)) such that A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)^(n+1)*x^n.

Original entry on oeis.org

1, 4, 11, 60, 611, 8632, 151538, 3132140, 73883667, 1949844168, 56785116742, 1806695366616, 62314198956510, 2315470815127792, 92214156916779444, 3918743752606940812, 177018691811732542595, 8471087431826716955880, 428141645771934036086942, 22791557465710675500959688
Offset: 0

Views

Author

Paul D. Hanna, May 21 2014

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 11*x^2 + 60*x^3 + 611*x^4 + 8632*x^5 + 151538*x^6 +...
such that A(x*G(x)) = G(x) where:
G(x) = 1 + 4*x + 27*x^2 + 256*x^3 + 3125*x^4 +...+ (n+1)^(n+1)*x^n +...
also, A(x) = G(x/A(x)):
A(x) = 1 + 4*x/A(x) + 27*x^2/A(x)^2 + 256*x^3/A(x)^3 + 3125*x^4/A(x)^4 +...+ (n+1)^(n+1)*x^n/A(x)^n +...
If we form a table of coefficients of x^k in A(x)^n like so:
[1,  4,  11,   60,   611,   8632,  151538,   3132140, ...];
[1,  8,  38,  208,  1823,  23472,  389174,   7739808, ...];
[1, 12,  81,  508,  4164,  48852,  759407,  14463624, ...];
[1, 16, 140, 1024,  8418,  91920, 1335712,  24248640, ...];
[1, 20, 215, 1820, 15625, 163664, 2232620,  38498580, ...];
[1, 24, 306, 2960, 27081, 279936, 3623894,  59297664, ...];
[1, 28, 413, 4508, 44338, 462476, 5764801,  89716400, ...];
[1, 32, 536, 6528, 69204, 739936, 9018480, 134217728, ...]; ...
then the main diagonal forms the sequence A007778:
[1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, ..., (n+1)^(n+2), ...].

Crossrefs

Cf. A180749, A007778.

Programs

  • PARI
    {a(n)=polcoeff(x/serreverse(x*sum(m=0, n+1, (m+1)^(m+1)*x^m)+x^2*O(x^n)), n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)^(n+2).
G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (n+1)^(n+1)*x^n.
Showing 1-3 of 3 results.

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