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.

A132070 G.f. satisfies: A(x) = 1 + x*A(x) * A(x*A(x)) + x^2*A(x)^2 * A'(x*A(x)).

Original entry on oeis.org

1, 1, 3, 16, 121, 1166, 13508, 181616, 2769085, 47109136, 883639920, 18107148225, 402374973506, 9637795235498, 247545497301618, 6787859682565805, 197933312666622029, 6116627434424525916, 199695744998600583128, 6868763166262716491823, 248281343442023732418598
Offset: 0

Views

Author

Paul D. Hanna, Aug 08 2007

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 121*x^4 + 1166*x^5 + 13508*x^6 + ...
Related expansions:
A(x*A(x)) = 1 + x + 4*x^2 + 25*x^3 + 206*x^4 + 2077*x^5 + 24603*x^6 + ...
A'(x*A(x)) = 1 + 6*x + 54*x^2 + 598*x^3 + 7714*x^4 + 112726*x^5 + ...
A(x)*A(A(x)) = 1 + 2*x + 8*x^2 + 48*x^3 + 380*x^4 + 3709*x^5 + 42856*x^6 + ...
A(x)^2*A'(A(x)) = 1 + 8*x + 73*x^2 + 786*x^3 + 9799*x^4 + 138760*x^5 + ...
where A(x) = 1 + x*A(x)*A(x*A(x)) + x^2*A(x)^2*A'(x*A(x)).
Let G(x) = 1 + x*A(x) + x^2*A'(x) be the g.f. of A113882;
the table of coefficients in G(x)^n begins:
G(x)^1: [(1), 1, 2, 9, 64, 605, 6996, 94556, ...];
G(x)^2: [1, (2), 5, 22, 150, 1374, 15539, 206676, ...];
G(x)^3: [1, 3, (9), 40, 264, 2346, 25937, 339294, ...];
G(x)^4: [1, 4, 14, (64), 413, 3568, 38558, 495848, ...];
G(x)^5: [1, 5, 20, 95, (605), 5096, 53840, 680365, ...];
G(x)^6: [1, 6, 27, 134, 849, (6996), 72302, 897558, ...];
G(x)^7: [1, 7, 35, 182, 1155, 9345, (94556), 1152936, ...]; ...
where the terms in parentheses form the initial terms of this sequence:
[(1)/1, (2)/2, (9)/3, (64)/4, (605)/5, (6996)/6, (94556)/7, ...].
Further, the logarithm of the g.f. A(x) may be formed from a diagonal in the above table:
log(A(x)) = x + 5*x^2/2 + 40*x^3/3 + 413*x^4/4 + 5096*x^5/5 + 72302*x^6/6 + 1152936*x^7/7 + ...

Crossrefs

Cf. A113882.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A*subst(A,x,x*A+x*O(x^n))+x^2*A^2*subst(A',x,x*A+x*O(x^n)));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    {a(n)=local(G=1+x);for(k=0,n,G=1+x*deriv(serreverse(x/(G+x^2*O(x^n))))); polcoeff(serreverse(x/G)/x,n)}

Formula

a(n) = A113882(n+1) / (n+1) for n>=0.
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x/(1 + x*A(x) + x^2*A'(x))).
(2) A(x) = (1/x)*Series_Reversion(x/G(x)) = G(x*A(x)) where G(x) = 1 + x*A(x) + x^2*A'(x) = A(x/G(x)) is the g.f. of A113882 (number of well-nested drawings of a rooted tree).

Extensions

More terms and name changed by Paul D. Hanna, Apr 28 2012

A361048 Expansion of g.f. A(x) satisfying a(n) = [x^(n-1)] A(x)^(n+1) for n >= 1.

Original entry on oeis.org

1, 1, 3, 18, 160, 1830, 25074, 395248, 6990876, 136464705, 2906040280, 66938704602, 1656963703434, 43848218457953, 1235194571623950, 36905133359883240, 1165832901366137184, 38830278855693956931, 1360186936717777641747, 49995325008141402758320
Offset: 0

Views

Author

Paul D. Hanna, Mar 13 2023

Keywords

Comments

A variant of A113882 (number of well-nested drawings of a rooted tree).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 160*x^4 + 1830*x^5 + 25074*x^6 + 395248*x^7 + 6990876*x^8 + 136464705*x^9 + ...
The table of coefficients in the successive powers of g.f. A(x) begins:
n = 1: [1, 1,  3,  18,  160,  1830,  25074,  395248, ...];
n = 2: [1, 2,  7,  42,  365,  4088,  55092,  857384, ...];
n = 3: [1, 3, 12,  73,  624,  6855,  90885, 1396236, ...];
n = 4: [1, 4, 18, 112,  947, 10224, 133410, 2023028, ...];
n = 5: [1, 5, 25, 160, 1345, 14301, 183765, 2750560, ...];
n = 6: [1, 6, 33, 218, 1830, 19206, 243205, 3593406, ...];
n = 7: [1, 7, 42, 287, 2415, 25074, 313159, 4568131, ...];
n = 8: [1, 8, 52, 368, 3114, 32056, 395248, 5693528, ...];
n = 9: [1, 9, 63, 462, 3942, 40320, 491304, 6990876, ...];
...
in which the secondary diagonal equals this sequence (shift left one position).
RELATION TO A361049.
The main diagonal in the above table,
[1, 2, 12, 112, 1345, 19206, 313159, 5693528, ...],
can be used to obtain the coefficients of the g.f. B(x) of A361049 like so:
[1, 2/2, 12/3, 112/4, 1345/5, 19206/6, 313159/7, 5693528/8, ...]
= [1, 1, 4, 28, 269, 3201, 44737, 711691, ..., A361049(n), ...].
Further, the coefficients in B(x)^2, which begins
[1, 2, 9, 64, 610, 7164, 98812, 1553528, 27292941, ...],
can in turn be used to generate the terms of this sequence:
[1*2/2, 2*3/2, 9*4/2, 64*5/2, 610*6/2, 7164*7/2, 98812*8/2, ...]
= [1, 3, 18, 160, 1830, 25074, 395248, ...].

Links

Crossrefs

Cf. A361049, A113882, A132070.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = -polcoeff( Ser(A) - x*Ser(A)^(#A), #A-1)); A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

Given g.f. A(x) = Sum_{n>=0} a(n)*x^n, let B(x) be the g.f. of A361049, then the following formulas hold.
(1) a(n) = [x^(n-1)] A(x)^(n+1) for n >= 1.
(2) a(n) = (n+1)/2 * [x^(n-1)] B(x)^2 for n >= 1.
(3) A(x) = 1 + x*B(x)^2 + x^2*B(x)*B'(x).
(4) A(x) = B(x/A(x)).
(5) A(x) = x / Series_Reversion(x*B(x)).
(6) B(x) = A(x*B(x)).
a(n) ~ c * n! * n^(3*LambertW(1) - 2 + 3/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.078464448259604971209... - Vaclav Kotesovec, Mar 13 2023

A361049 G.f. satisfies: A(x) = (1/x)*Series_Reversion( x/(1 + x*A(x)^2 + x^2*A(x)*A'(x)) ).

Original entry on oeis.org

1, 1, 4, 28, 269, 3201, 44737, 711691, 12630023, 246594988, 5244025502, 120540052304, 2976918491501, 78601791684495, 2209667973082374, 65901745111752843, 2078619947109354811, 69141776287740239348, 2419303138068147399700, 88842295496847889690405
Offset: 0

Views

Author

Paul D. Hanna, Mar 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 269*x^4 + 3201*x^5 + 44737*x^6 + 711691*x^7 + 12630023*x^8 + 246594988*x^9 + ...
such that A(x) = G(x*A(x)) where G(x) is given by
G(x) = 1 + d/dx (x^2 * A(x)^2)/2, which begins
G(x) = 1 + x + 3*x^2 + 18*x^3 + 160*x^4 + 1830*x^5 + 25074*x^6 + 395248*x^7 + 6990876*x^8 + ... + A361048(n)*x^n + ...

Links

Crossrefs

Cf. A361048, A132070, A113882.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n,
A = (1/x)*serreverse( x/(1 + x*A^2 + x^2*A*A') +x^2*O(x^n) )); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

Given g.f. A(x) = Sum_{n>=0} a(n)*x^n, let G(x) be the g.f. of A361048, then the following formulas hold.
(1) A(x) = (1/x)*Series_Reversion( x/(1 + x*A(x)^2 + x^2*A(x)*A'(x)) ).
(2) A(x) = G(x*A(x)).
(4) A(x) = (1/x)*Series_Reversion(x/G(x)).
(3) G(x) = A(x/G(x)).
(5) G(x) = 1 + d/dx (x^2 * A(x)^2)/2.
a(n) ~ c * n! * n^(3*LambertW(1) - 2 + 3/(1 + LambertW(1))) / LambertW(1)^n, where c = 0.13835030685615842626... - Vaclav Kotesovec, Mar 13 2023
Showing 1-3 of 3 results.

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