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.

Previous Showing 11-16 of 16 results.

A191807 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(n^5).

Original entry on oeis.org

1, 1, 2, 35, 839, 39817, 3186197, 350165095, 60551444146, 14198688038403, 4371837596384520, 1810022601411867232, 944544112975598084882, 612407529661033348265928, 494617108638856876638563800
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 35*x^3 + 839*x^4 + 39817*x^5 + 3186197*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^32 + x^3*A(x)^243 + x^4*A(x)^1024 + x^5*A(x)^3125 + x^6*A(x)^7776 +...+ x^n*A(x)^(n^5) +...

Crossrefs

Cf. A107595, A191805, A191806, A191808.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(m^5)));polcoeff(A,n)}

A191808 G.f. satisfies: A(x) = Sum_{n>=0} x^n*A(x)^(n^6).

Original entry on oeis.org

1, 1, 2, 67, 2941, 327868, 75504909, 23827165139, 14250056009694, 12425294009958329, 14645969719539739213, 26039632087532302145593, 61861506372519998418244631, 188885323588779065057458593218
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 67*x^3 + 2941*x^4 + 327868*x^5 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^64 + x^3*A(x)^729 + x^4*A(x)^4096 + x^5*A(x)^15625 + x^6*A(x)^46656 +...+ x^n*A(x)^(n^6) +...

Crossrefs

Cf. A107595, A191805, A191806, A191807.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(m^6)));polcoeff(A,n)}

A107594 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2-n).

Original entry on oeis.org

1, 1, 1, 3, 10, 42, 194, 979, 5274, 30037, 179527, 1120612, 7280750, 49120810, 343547469, 2487670468, 18631824735, 144215785791, 1152745117570, 9508011730755, 80861962283808, 708502494881786, 6390084112199801, 59272034375915217, 564899767969587670
Offset: 0

Views

Author

Paul D. Hanna, May 17 2005

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 42*x^5 + 194*x^6 + 979*x^7 +...
Let A = A(x) then
A = 1 + x*A^0 + x^2*A^2 + x^3*A^6 + x^4*A^12 + x^5*A^20 + x^6*A^30 +...
= 1 + x + (x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 27*x^6 + 110*x^7 +...)
+ (x^3 + 6*x^4 + 21*x^5 + 68*x^6 + 240*x^7 + 948*x^8 + 4140*x^9 +...)
+ (x^4 + 12*x^5 + 78*x^6 + 388*x^7 + 1737*x^8 + 7632*x^9 +...)
+ (x^5 + 20*x^6 + 210*x^7 + 1580*x^8 + 9795*x^9 +...)
+ (x^6 + 30*x^7 + 465*x^8 + 5020*x^9 +...) +...

Crossrefs

Cf. A107590, A107595, A107596.

Programs

  • Mathematica
    m = 25; A[_] = 0;
Do[A[x_] = 1 + x + Sum[x^k A[x]^(k^2 - k) + O[x]^j, {k, 2, j}], {j, m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 05 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(k=1,n,A=1+sum(j=1,n,x^j*A^(j^2-j)+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = x/series-reversion(x*G107595(x)) and thus A(x) = G107595(x/A(x)) where G107595(x) is the g.f. of A107595.
G.f. A(x)^2 = x/series-reversion(x*G107596(x)^2) and thus A(x) = G107596(x/A(x)^2) where G107596(x) is the g.f. of A107596.
From Paul D. Hanna, Apr 25 2010: (Start)
Let A = g.f. A(x), then A satisfies the continued fraction:
A = 1/(1- x/(1- (A^2-1)*x/(1- A^4*x/(1- (A^6-A^2)*x/(1- A^8*x/(1- (A^10-A^4)*x/(1- A^12*x/(1- (A^14-A^6)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)

A107596 G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2+n).

Original entry on oeis.org

1, 1, 3, 14, 80, 514, 3567, 26153, 199900, 1579107, 12816020, 106421359, 901430144, 7771535382, 68085001080, 605420138920, 5459655601753, 49904765136264, 462228258349278, 4337787743946224, 41249375376404380, 397572319756235577
Offset: 0

Views

Author

Paul D. Hanna, May 17 2005

Keywords

Examples

			A = 1 + x*A^2 + x^2*A^6 + x^3*A^12 + x^4*A^20 + x^5*A^30 ...
= 1 + (x + 2*x^2 + 7*x^3 + 34*x^4 + 197*x^5 + 1272*x^6 +...)
+ (x^2 + 6*x^3 + 33*x^4 + 194*x^5 + 1230*x^6 +...)
+ (x^3 + 12*x^4 + 102*x^5 + 784*x^6 +...)
+ (x^4 + 20*x^5 + 250*x^6 +...) +...
= 1 + x + 3*x^2 + 14*x^3 + 80*x^4 + 514*x^5 + 3567*x^6 +...

Crossrefs

Cf. A107592, A107594, A107595.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(k=1,n,A=1+sum(j=1,n,x^j*A^(j^2+j)+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. A(x)^2 = (1/x)*series-reversion(x/G107594(x)^2) and thus A(x) = G107594(x*A(x)^2) where G107594(x) is the g.f. of A107594. G.f. A(x) = (1/x)*series-reversion(x/G107595(x)) and thus A(x) = G107595(x*A(x)) where G107595(x) is the g.f. of A107595.
Contribution from Paul D. Hanna, Apr 25 2010: (Start)
Let A = g.f. A(x), then A satisfies the continued fraction:
A = 1/(1- A^2*x/(1- (A^4-A^2)*x/(1- A^6*x/(1- (A^8-A^4)*x/(1- A^10*x/(1- (A^12-A^6)*x/(1- A^14*x/(1- (A^16-A^8)*x/(1- A^18*x)))))))))
due to an identity of a partial elliptic theta function.
(End)

A234855 E.g.f. satisfies: A(x) = Sum_{n>=0} ( Integral A(x)^n dx )^n.

Original entry on oeis.org

1, 1, 3, 21, 241, 3951, 85499, 2325205, 76860673, 3014656183, 137784836475, 7235668490589, 431692029451009, 28991550501283359, 2174713803535479419, 181001542259074421413, 16618721538838243841185, 1674634828088234390862727, 184352162064651888588105243
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 241*x^4/4! + 3951*x^5/5! +...
To illustrate how the terms are generated, form a table of coefficients of x^k/k!, k>=0, in (Integral A(x)^n dx)^n for n>=0 like so:
n=0: [1, 0, 0,  0,   0,    0,     0,      0,        0,         0, ...];
n=1: [0, 1, 1,  3,  21,  241,  3951,  85499,  2325205,  76860673, ...];
n=2: [0, 0, 2, 12,  88,  920, 13328, 254744,  6161568, 182632584, ...];
n=3: [0, 0, 0,  6, 108, 1710, 29700, 600642, 14344092, 403670790, ...];
n=4: [0, 0, 0,  0,  24,  960, 28800, 826560, 24665088, 793449216, ...];
n=5: [0, 0, 0,  0,   0,  120,  9000, 462000, 20958000, 922005000, ...];
n=6: [0, 0, 0,  0,   0,    0,   720,  90720,  7378560, 504040320, ...];
n=7: [0, 0, 0,  0,   0,    0,     0,   5040,   987840, 120022560, ...];
n=8: [0, 0, 0,  0,   0,    0,     0,      0,    40320,  11612160, ...];
n=9: [0, 0, 0,  0,   0,    0,     0,      0,        0,    362880, ...]; ...
then the column sums form the terms of this sequence.

Links

Crossrefs

Cf. A232552, A107595.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(k=0,30,intformal( (A+x*O(x^n))^k )^k));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. satisfies: A'(x) = Sum_{n>=1} n * A(x)^n * ( Integral A(x)^n dx )^(n-1).

A192260 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + A(x))^(2*n) * A(x)^(n^2).

Original entry on oeis.org

1, 4, 48, 912, 21184, 552320, 15532032, 460947712, 14247537664, 454761822208, 14902431522816, 499315007266816, 17054726818791424, 592541668923539456, 20907267781281054720, 748286964823747526656, 27143591551031801806848, 27143591551031801806848, 997356616630147913089024, 37108619649604340227768320, 1397931208210552892111716352, 53322215792785853528148017152, 2059866344459108561028558880768, 80619871370319975775336625340416
Offset: 0

Views

Author

Paul D. Hanna, Jun 26 2011

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 48*x^2 + 912*x^3 + 21184*x^4 + 552320*x^5 +...
Let A = g.f. A(x), then A satisfies:
A = 1 + x*(1+A)^2*A + x^2*(1+A)^4*A^4 + x^3*(1+A)^6*A^9 + x^4*(1+A)^8*A^16 +...
Equivalently,
A = 1 + x*(A + 2*A^2 + A^3) + x^2*(A^4 + 4*A^5 + 6*A^6 + 4*A^7 + A^8) + x^3*(A^9 + 6*A^10 + 15*A^11 + 20*A^12 + 15*A^13 + 6*A^14 + A^15) +...

Crossrefs

Cf. A107595, A192259.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(1+A)^(2*m)*(A+x*O(x^n))^(m^2)));polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

Formula

Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*(1+A)^(2*n)*A^n * Product_{k=1..n} (1 - x*(1+A)^2*A^(4*k-3))/(1 - x*(1+A)^2*A^(4*k-1))
(2) A = 1/(1- A*(1+A)^2*x/(1- A*(A^2-1)*(1+A)^2*x/(1- A^5*(1+A)^2*x/(1- A^3*(A^4-1)*(1+A)^2*x/(1- A^9*(1+A)^2*x/(1- A^5*(A^6-1)*(1+A)^2*x/(1- A^13*(1+A)^2*x/(1- A^7*(A^8-1)*(1+A)^2*x/(1- ...))))))))) (continued fraction).
The above formulas are due to (1) a q-series identity and (2) a partial elliptic theta function expression.
Previous Showing 11-16 of 16 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.