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 1-4 of 4 results.

A095793 G.f.: A(x) = 1+x*(1+x*(1+x*(...(1+x*(...)^n)...)^3)^2)^1.

Original entry on oeis.org

1, 1, 1, 2, 7, 36, 245, 2072, 20913, 245012, 3265581, 48766020, 806254126, 14616629622, 288272307999, 6144034279588, 140715744051270, 3446290524236454, 89874216926157157, 2486386071747194244
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2004

Keywords

Examples

			G.f.: 1 + x + x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 245*x^6 + 2072*x^7 +...
such that
A(x) = 1 + x*B(x), B(x) = 1 + x*C(x)^2, C(x) = 1 + x*D(x)^3, D(x) = 1 + x*E(x)^4, E(x) = 1 + x*F(x)^5, F(x) = 1 + x*G(x)^6, G(x) = 1 + x*H(x)^7, ...
where A(x), B(x), C(x), ... are the g.f. of the sequences given below.
A: [1, 1, 1, 2, 7, 36, 245, 2072, 20913, 245012, ...];
B: [1, 1, 2, 7, 36, 245, 2072, 20913, 245012, 3265581, ...];
C: [1, 1, 3, 15, 103, 888, 9147, 109150, 1477575, 22349316, ...];
D: [1, 1, 4, 26, 224, 2351, 28760, 399314, 6183132, 105455687, ...];
E: [1, 1, 5, 40, 415, 5145, 73121, 1162620, 20358145, 388334030, ...];
F: [1, 1, 6, 57, 692, 9906, 160656, 2884554, 56502264, 1195386975, ...];
G: [1, 1, 7, 77, 1071, 17395, 317303, 6357267, 137950303, 3211604480, ...];
H: [1, 1, 8, 100, 1568, 28498, 577808, 12788776, 304827080, 7753676623, ...];
I: [1, 1, 9, 126, 2199, 44226, 987021, 23928972, 621887265, 17173176273, ...]; ...
FIRST DERIVATIVES OF SERIES:
A' = B + x*C^2 + 2!*x^2*C*D^3 + 3!*x^3*C*D^2*E^4 + 4!*x^4*C*D^2*E^3*F^5 + 5!*x^5*C*D^2*E^3*F^4*G^6 + 6!*x^6*C*D^2*E^3*F^4*G^5*H^7 +...
B' = C^2 + 2!*x*C*D^3 + 3!*x^2*C*D^2*E^4 + 4!*x^3*C*D^2*E^3*F^5 + 5!*x^4*C*D^2*E^3*F^4*G^6 + 6!*x^5*C*D^2*E^3*F^4*G^5*H^7 +...
2!*C' = 2!*D^3 + 3!*x*D^2*E^4 + 4!*x^2*D^2*E^3*F^5 + 5!*x^3*D^2*E^3*F^4*G^6 + 6!*x^4*D^2*E^3*F^4*G^5*H^7 + 7!*x^5*D^2*E^3*F^4*G^5*H^6*I^8 +...

Links

Crossrefs

Cf. A096537, A234301.

Programs

  • PARI
    {a(n)=local(A);A=1+x+x*O(x^n);for(j=0,n-1,A=1+x*A^(n-j));polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Print Row r in Table (this Sequence is at r=1) */
{a(n,r=1)=local(A=vector(3*n+2*r+2,i,1+x));
for(m=1,2*n+r,for(j=0,n+r+m, A[n+r+m-j+1]=1+x*(A[n+r+m-j+2] +x^r*O(x^n))^(n+r+m-j+1) ););polcoeff(A[r],n)}
for(n=0, 20, print1(a(n,1), ", "))

A096538 E.g.f.: A(x) = exp(x*exp(2*x*exp(2^2*x*exp(...exp(2^n*x*exp(...))...)))), for n>=0.

Original entry on oeis.org

1, 1, 5, 73, 2649, 226881, 45061213, 20520985353, 21182201493617, 48996888022427329, 251357040234734546421, 2834058902388354210737289, 69683890614563169975467620681, 3711434364793976039520825570430593
Offset: 0

Views

Author

Paul D. Hanna, Jun 24 2004

Keywords

Examples

			A(x) = 1 + 1*x + 5*x^2/2! + 73*x^3/3! + 2649*x^4/4! + 226881*x^5/5! +...

Links

Crossrefs

Cf. A096537.

Programs

  • PARI
    a(n)=local(A=exp(x));for(i=1,n,A=exp(x*(2^(n-i))*A+x*O(x^n)));n!*polcoeff(A,n)

Formula

E.g.f. satisfies: log(A(x)) = x*A(2*x).
a(n+1) = Sum_{i=0..n} (i+1)*2^i*binomial(n,i)*a(i)*a(n-i). - Vladeta Jovovic, Dec 29 2006
a(n) ~ c * n! * 2^(n*(n-1)/2), where c = 1.972549257529822552687919986141209749606505056... . - Vaclav Kotesovec, Jul 31 2014

A096542 Triangle, read by rows, where e.g.f. A(x,y) satisfies: A(x,y) = exp(x*y*A(x,y+1)) and A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k)/n!*x^n*y^k.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 15, 30, 16, 0, 244, 564, 444, 125, 0, 6885, 17540, 16680, 7320, 1296, 0, 298326, 817470, 877740, 478380, 136590, 16807, 0, 18377191, 53352138, 62582100, 39142600, 14146440, 2873136, 262144, 0, 1525885992, 4645224472
Offset: 0

Views

Author

Paul D. Hanna, Jun 25 2004

Keywords

Comments

Row sums form A096537.
Main diagonal forms A000272 (labeled trees on n nodes).
Secondary diagonal forms 2*A057500 (labeled connected graphs with n edges and n nodes).
Other diagonals include 3*A096543 and 4*A096544.

Examples

			A(x,y) = exp(x*y*exp(x*(y+1)*exp(x*(y+2)*exp(...exp(x*(n+y)*exp(...))...)))).
Triangle begins:
1;
0, 1;
0, 2, 3;
0, 15, 30, 16;
0, 244, 564, 444, 125;
0, 6885, 17540, 16680, 7320, 1296;
0, 298326, 817470, 877740, 478380, 136590, 16807;
0, 18377191, 53352138, 62582100, 39142600, 14146440, 2873136, 262144;
0, 1525885992, 4645224472, 5837707848, 4032207480, 1692155640, 441093240, 67558680, 4782969; ...

Links

Crossrefs

Cf. A096537, A000272, A057500, A096543, A096544.

Programs

  • PARI
    {T(n,k)=local(A=exp(x));for(i=1,n,A=exp(x*(n-i+y)*A+x*O(x^n)+y*O(y^k))); n!*polcoeff(polcoeff(A,k,y),n,x)}

Formula

E.g.f. satisfies: A(x, y+1) = log(A(x, y))/(x*y).
T(n, 1) = n*A096537(n).
T(n, n) = (n+1)^(n-1) = A000272(n+1).
T(n, n-1) = 2*A057500(n).

A189897 E.g.f.: A(x) = exp(x*exp(x*exp(2*x*exp(3*x*exp(...exp(n*x*exp(...))...))))).

Original entry on oeis.org

1, 1, 3, 22, 329, 8636, 355297, 21117286, 1710243761, 180811765432, 24158025584801, 3977274470362634, 790696358461658761, 186695449895152470052, 51635196859642278380513, 16532803795918313120452246
Offset: 0

Views

Author

Paul D. Hanna, May 01 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 329*x^4/4! + 8636*x^5/5! +...
The e.g.f. and related series satisfy:
A(x) = exp(x*B), B = exp(x*C^2), C = exp(x*D^3), D = exp(x*E^4), E = exp(x*F^5), F = exp(x*G^6), ...
where the series begin:
B = 1 + x + 5*x^2/2! + 61*x^3/3! + 1377*x^4/4! + 49721*x^5/5! +...
C = 1 + x + 7*x^2/2! + 118*x^3/3! + 3529*x^4/4! + 162076*x^5/5! +...
D = 1 + x + 9*x^2/2! + 193*x^3/3! + 7169*x^4/4! + 399521*x^5/5! +...
E = 1 + x + 11*x^2/2! + 286*x^3/3! + 12681*x^4/4! + 830876*x^5/5! +...
F = 1 + x + 13*x^2/2! + 397*x^3/3! + 20449*x^4/4! + 1539961*x^5/5! +...
G = 1 + x + 15*x^2/2! + 526*x^3/3! + 30857*x^4/4! + 2625596*x^5/5! +...
Relevant powers of the above series begin:
C^2 = 1 + 2*x + 16*x^2/2! + 278*x^3/3! + 8296*x^4/4! + 375962*x^5/5! +...
D^3 = 1 + 3*x + 33*x^2/2! + 747*x^3/3! + 27921*x^4/4! + 1536723*x^5/5! +...
E^4 = 1 + 4*x + 56*x^2/2! + 1564*x^3/3! + 70416*x^4/4! + 4576724*x^5/5! +...
F^5 = 1 + 5*x + 85*x^2/2! + 2825*x^3/3! + 148945*x^4/4! + 11182925*x^5/5! +...
G^6 = 1 + 6*x + 120*x^2/2! + 4626*x^3/3! + 279672*x^4/4! + 23840046*x^5/5! +...

Links

Crossrefs

Cf. A096537.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(x*(n-i+1)*A+x*O(x^n))); n!*polcoeff(exp(x*A), n)}

Formula

E.g.f.: A(x) = exp(x*B(x)) where B(x) is the e.g.f. of A096537.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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