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 41-44 of 44 results.

A242795 a(n) = [x^n] ( 1 + x*A(x)^(n+1) )^(n+1) / (n+1) for n>=0, with a(0)=1.

Original entry on oeis.org

1, 1, 4, 35, 466, 8072, 168330, 4045603, 109089538, 3242538284, 104946776716, 3665946814257, 137291732981170, 5483948111154008, 232660368810666229, 10447887814670412307, 495139557493903545618, 24699919104193662112382, 1293980412831022073519196
Offset: 0

Views

Author

Paul D. Hanna, May 22 2014

Keywords

Comments

Compare to the g.f. G(x) = x + x*G(G(x)) of A030266 that satisfies:
A030266(n+1) = [x^n] ( 1 + G(x) )^(n+1) / (n+1) for n>=0.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 35*x^3 + 466*x^4 + 8072*x^5 + 168330*x^6 +...
Form a table of coefficients of x^k in (1 + x*A(x)^(n+1))^(n+1) like so:
n=0: [1,  1,   1,    4,    35,    466,    8072,   168330, ...];
n=1: [1,  2,   5,   22,   178,   2228,   37141,   755702, ...];
n=2: [1,  3,  12,   64,   516,   6126,   97725,  1929927, ...];
n=3: [1,  4,  22,  140,  1177,  13548,  206876,  3946612, ...];
n=4: [1,  5,  35,  260,  2330,  26626,  391830,  7202170, ...];
n=5: [1,  6,  51,  434,  4185,  48432,  694714, 12312642, ...];
n=6: [1,  7,  70,  672,  6993,  83174, 1178310, 20224653, ...];
n=7: [1,  8,  92,  984, 11046, 136392, 1932876, 32364824, ...];
n=8: [1,  9, 117, 1380, 16677, 215154, 3084024, 50833962, ...];
n=9: [1, 10, 145, 1870, 24260, 328252, 4801655, 78652350, ...]; ...
then this sequence is formed from the main diagonal:
[1/1, 2/2, 12/3, 140/4, 2330/5, 48432/6, 1178310/7, 32364824/8, ...].

Crossrefs

Cf. A242794.

Programs

  • PARI
    {a(n)=local(A=[1,1]);for(m=1,n,A=concat(A,0);A[m+1]=Vec((1+x*Ser(A)^(m+1))^(m+1))[m+1]/(m+1));A[n+1]}
for(n=0,25,print1(a(n),", "))

A247224 G.f. A(x) satisfies: A(x) = B(x)*(A(x) - x*C(x)) where B(x) = A(x/B(x)) and C(x) = A(x*C(x)).

Original entry on oeis.org

1, 1, 2, 6, 26, 149, 1024, 7965, 68192, 632724, 6294190, 66579501, 744194484, 8747497833, 107718981328, 1385436413289, 18563761993762, 258579817821799, 3737335096804136, 55957031888334621, 866632465992896412, 13865193902724224273, 228875892203793317404, 3893773927147402337094
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2014

Keywords

Comments

Compare to: G(x) = Series_Reversion( x - Series_Reversion(x*G(x)) * x*G(x) )/x, which is satisfied by the g.f. G(x) = 1 + x*G(x) * G(x*G(x)) of A030266 with offset 0.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 149*x^5 + 1024*x^6 +...
Let B(x) = A(x/B(x)) and C(x) = A(x*C(x)), where B(x) and C(x) begin:
B(x) = 1 + x + x^2 + 2*x^3 + 9*x^4 + 53*x^5 + 354*x^6 + 2651*x^7 + 21951*x^8 + 197666*x^9 + 1911091*x^10 + 19665622*x^11 + 214060860*x^12 +...
C(x) = 1 + x + 3*x^2 + 13*x^3 + 71*x^4 + 460*x^5 + 3399*x^6 + 27867*x^7 + 248789*x^8 + 2388199*x^9 + 24432778*x^10 + 264682253*x^11 + 3021086129*x^12 +...
then A(x) = B(x) * (A(x) - x*C(x)).

Links

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion( x - Series_Reversion(x/A(x)) * x/A(x) )/x.
(2) A(x) = x/Series_Reversion( (x - Series_Reversion(x*A(x))) * A(x)/x ).
Given B(x) = A(x/B(x)) and C(x) = A(x*C(x)), then:
(3.a) A(x) = B(x*A(x)) and A(x) = C(x/A(x)),
(3.b) B(x) = x/Series_Reversion(x*A(x)),
(3.c) C(x) = Series_Reversion(x/A(x))/x,
(3.d) B(x) = A(x)/(A(x) - x*C(x)),
(3.e) C(x) = A(x)*(1 - 1/B(x))/x.

A378576 G.f. satisfies A(x) = x + x*A(A(A(A(A(A(x)))))), so that this sequence shifts left under the 6th self-COMPOSE.

Original entry on oeis.org

1, 1, 6, 66, 981, 17576, 359101, 8109026, 198480901, 5197916551, 144326504376, 4220683214771, 129349023338616, 4138098976882836, 137770738524681831, 4761510500867829696, 170476220596490911691, 6311806909067054474161, 241301669599996597349506, 9512867537981387958217696, 386276838889195561353811321
Offset: 1

Views

Author

Paul D. Hanna, Dec 01 2024

Keywords

Comments

Conjecture: a(n) == 1 (mod 5) for n >= 1.

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 66*x^4 + 981*x^5 + 17576*x^6 + 359101*x^7 + 8109026*x^8 + 198480901*x^9 + 5197916551*x^10 + ...
where A(x) = x + x*A^6(x).
RELATED SERIES.
A^2(x) = A(A(x)) = x + 2*x^2 + 14*x^3 + 163*x^4 + 2496*x^5 + 45577*x^6 + 944034*x^7 + 21546983*x^8 + 532083781*x^9 + 14041137012*x^10 + ...
A^3(x) = A(A(A(x))) = x + 3*x^2 + 24*x^3 + 297*x^4 + 4711*x^5 + 88073*x^6 + 1856179*x^7 + 42949757*x^8 + 1072699696*x^9 + 28584771073*x^10 + ...
A^4(x) = A(A(A(A(x)))) = x + 4*x^2 + 36*x^3 + 474*x^4 + 7816*x^5 + 150144*x^6 + 3230016*x^7 + 75976854*x^8 + 1923771726*x^9 + 51873235544*x^10 + ...
A^5(x) = A(A(A(A(A(x))))) = x + 5*x^2 + 50*x^3 + 700*x^4 + 12025*x^5 + 238000*x^6 + 5240145*x^7 + 125605650*x^8 + 3231333675*x^9 + 88338524825*x^10 + ...
...
A^6(x) = A(A(A(A(A(A(x)))))) = x + 6*x^2 + 66*x^3 + 981*x^4 + 17576*x^5 + 359101*x^6 + 8109026*x^7 + 198480901*x^8 + 5197916551*x^9 + 144326504376*x^10 + ...
By formula (4),
A(x) = x + x*A^5(x) + x*A^5(x)*A^10(x) + x*A^5(x)*A^10(x)*A^15(x) + x*A^5(x)*A^10(x)*A^15(x)*A^20(x) + ...
Examples of formula (5), A^n(x) = A^(n+1)(x)/(1 + A^(n+6)(x)):
n=0: x = A(x)/(1 + A(A(A(A(A(A(x))))))),
n=1: A(x) = A(A(x))/(1 + A(A(A(A(A(A(A(x)))))))),
n=2: A(A(x)) = A(A(A(x)))/(1 + A(A(A(A(A(A(A(A(x))))))))),
n=3: A(A(A(x))) = A(A(A(A(x))))/(1 + A(A(A(A(A(A(A(A(A(x)))))))))),
...
Examples of formula (6), A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+6)(x)):
n=1: A(x) = x*(1 + A(A(A(A(A(A(x))))))),
n=2: A(A(x)) = x*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x)))))))),
n=3: A(A(A(x))) = x*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x))))))))*(1 + A(A(A(A(A(A(A(A(x))))))))),
...

Links

Crossrefs

Cf. A030266, A091713, A196523, A378575.

Programs

  • PARI
    /* By definition, A(x) = x + x*A(A(A(A(A(A(x)))))) */
/* Define the n-th iteration of function F: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
{a(n) = my(A=x); for(i=1, n, A = x + x*ITERATE(6, A, n)); polcoef(A, n)}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas, wherein A^n(x) denotes the n-th iteration of A(x) with A^0(x) = x.
(1) A(x) = x + x*A^6(x).
(2) A(x) = A(A(x))/(1 + A^7(x)).
(3) A(x) = Series_Reversion( x/(1 + A^5(x)) ).
(4) A(x) = Sum_{n>=0} Product_{k=0..n} A^(5*k)(x).
(5) A^n(x) = A^(n+1)(x) / (1 + A^(n+6)(x)) for n >= 0.
(6) A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+6)(x)) for n >= 1.

A187820 G.f. satisfies: A(x) = x + x*Sum{n>=0} x^n/n! * d^n/dx^n A(x)^(n+1).

Original entry on oeis.org

1, 1, 3, 12, 62, 377, 2585, 19497, 159113, 1389122, 12868421, 125721997, 1289307566, 13827494435, 154617546534, 1798091282057, 21700908011943, 271305769072192, 3507957527328398, 46842455163101344, 645137348798871851, 9153330597244491848, 133643964236921732563
Offset: 1

Views

Author

Paul D. Hanna, Dec 27 2012

Keywords

Comments

Compare to g.f. G(x) = x + x*G(G(x)) of A030266, where:
G(x) = x + x*Sum{n>=0} x^n/n! * d^n/dx^n G(x)^(n+1)/(n+1).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 62*x^5 + 377*x^6 + 2585*x^7 +...
where
A(x) = x + x*A(x)/0! + x^2*d/dx A(x)^2/1! + x^3*d^2/dx^2 A(x)^3/2! + x^4*d^3/dx^3 A(x)^4/3! + x^5*d^4/dx^4 A(x)^5/4! +...

Crossrefs

Cf. A030266.

Programs

  • PARI
    {Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D}
{a(n)=local(A=x+x*O(x^n)); for(i=1,n,A=x+x*sum(m=0,n,x^m/m!*Dx(m,(A+x*O(x^n))^(m+1)) )); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
Previous Showing 41-44 of 44 results.

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