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.

A128318 G.f.: A(x) = 1+x*(1+2x*(1+3x*(...(1+n*x*(...)^2)^2...)^2)^2)^2.

Original entry on oeis.org

1, 1, 4, 28, 276, 3480, 53232, 955524, 19672320, 456803328, 11810032896, 336463895808, 10473959755008, 353739038360832, 12883270796310528, 503352328766459904, 21001144899441162240, 931963581151516477440, 43832663421577452887040, 2178029362561822117094400, 114014865901176834809333760
Offset: 0

Views

Author

Paul D. Hanna, Mar 07 2007

Keywords

Examples

			G.f.: A(x) = 1 + x*B(x)^2; B(x) = 1 + 2*x*C(x)^2; C(x) = 1 + 3*x*D(x)^2; D(x) = 1 + 4*x*E(x)^2; E(x) = 1 + 5*x*F(x)^2; F(x) = 1 + 6*x*G(x)^2; ...
where the respective sequences begin:
A=[1,1,4,28,276,3480,53232,955524,19672320,...];
B=[1,2,12,114,1440,22368,409248,8585088,202733760,...];
C=[1,3,24,288,4440,82080,1752000,42178800,1127335680,...];
D=[1,4,40,580,10560,226560,5532960,150570240,4501422240,...];
E=[1,5,60,1020,21420,523320,14399280,437433780,14479664640,...];
F=[1,6,84,1638,38976,1068480,32716992,1098069504,39896236800,...];
G=[1,7,112,2464,65520,1991808,67189248,2469837888,97765355520,...];
H=[1,8,144,3528,103680,3461760,127569600,5098406400,218459165760,...];

Links

Crossrefs

Cf. A128319, A268652.

Programs

  • PARI
    {a(n)=local(A=1+(n+1)*x);for(k=0,n,A=1+(n-k+1)*x*A^2 +x*O(x^n));polcoeff(A,n)}
for(n=0, 25, print1(a(n), ", "))

Formula

Conjecture: a(n) ~ n! * (8/3)^n / sqrt(n). - Vaclav Kotesovec, Mar 19 2016

A128570 Rectangular table, read by antidiagonals, where the g.f. of row n, R(x,n), satisfies: R(x,n) = 1 + (n+1)*x*R(x,n+1)^2 for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 12, 28, 1, 4, 24, 114, 276, 1, 5, 40, 288, 1440, 3480, 1, 6, 60, 580, 4440, 22368, 53232, 1, 7, 84, 1020, 10560, 82080, 409248, 955524, 1, 8, 112, 1638, 21420, 226560, 1752000, 8585088, 19672320, 1, 9, 144, 2464, 38976, 523320, 5532960, 42178800, 202733760, 456803328, 1, 10, 180, 3528, 65520, 1068480, 14399280, 150570240, 1127335680, 5317663680, 11810032896, 1, 11, 220, 4860, 103680, 1991808, 32716992, 437433780, 4501422240, 33073099200, 153345634560, 336463895808
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Comments

Row r > 0 is asymptotic to 2^(2*r) * n^r * A128318(n) / (3^r * r!). - Vaclav Kotesovec, Mar 19 2016

Examples

			Row g.f.s satisfy: R(x,0) = 1 + x*R(x,1)^2, R(x,1) = 1 + 2x*R(x,2)^2,
R(x,2) = 1 + 3x*R(x,3)^2, R(x,3) = 1 + 4x*R(x,4)^2, ...
where the initial rows begin:
R(x,0):[1,1,4,28,276,3480,53232,955524,19672320,456803328,...];
R(x,1):[1,2,12,114,1440,22368,409248,8585088,202733760,...];
R(x,2):[1,3,24,288,4440,82080,1752000,42178800,1127335680,...];
R(x,3):[1,4,40,580,10560,226560,5532960,150570240,4501422240,...];
R(x,4):[1,5,60,1020,21420,523320,14399280,437433780,14479664640,...];
R(x,5):[1,6,84,1638,38976,1068480,32716992,1098069504,39896236800,...];
R(x,6):[1,7,112,2464,65520,1991808,67189248,2469837888,97765355520,..];
R(x,7):[1,8,144,3528,103680,3461760,127569600,5098406400,...];
R(x,8):[1,9,180,4860,156420,5690520,227470320,9821970180,...];
R(x,9):[1,10,220,6490,227040,8939040,385265760,17875608960,..].

Links

Crossrefs

Rows: A128318, A128571, A128572, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).
Cf A268652.

Programs

  • PARI
    {T(n,k)=local(A=1+(n+k+1)*x); for(j=0,k,A=1+(n+k+1-j)*x*A^2 +x*O(x^k));polcoeff(A,k)}
for(n=0, 12, for(k=0, 10, print1(T(n, k), ", ")); print(""))

A128571 Row 1 of table A128570.

Original entry on oeis.org

1, 2, 12, 114, 1440, 22368, 409248, 8585088, 202733760, 5317663680, 153345634560, 4821848409600, 164211751261440, 6022162697840640, 236652023784960000, 9921992082873223680, 442138176056374548480, 20869300232695599552000, 1040210006521640127367680, 54600929159270409876879360
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Links

Crossrefs

Cf. A128570 (triangle), other rows: A128318, A128572, A128573, A128574, A128575, A128576; A128577 (square of row 0), A128578 (main diagonal), A128579 (antidiagonal sums).
Cf. A268652.

Programs

  • PARI
    {a(n)=local(A=1+(n+2)*x);for(j=0,n,A=1+(n+2-j)*x*A^2 +x*O(x^n)); polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: A(x) = 1 + 2x*R(x,2)^2, where R(x,2) = 1 + 3*x*R(x,3)^2, R(x,3) = 1 + 4*x*R(x,4)^2, ..., R(x,n) = 1 + (n+1)*x*R(x,n+1)^2, ... and R(x,n) is the g.f. of row n of table A128570.
a(n) ~ 4*n*A128318(n)/3. - Vaclav Kotesovec, Mar 19 2016
Showing 1-3 of 3 results.

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