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.

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A308030 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * (1 + A(A(x))).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 18, 45, 122, 350, 1052, 3313, 10933, 37739, 135865, 508545, 1973717, 7926795, 32895354, 140894024, 622160220, 2829323210, 13235526027, 63620528705, 313909404040, 1588405927920, 8235905545581, 43724990832997, 237527663672208, 1319398402129845
Offset: 1

Views

Author

Ilya Gutkovskiy, May 10 2019

Keywords

Comments

Shifts left 3 places under COMPOSE transform.

Crossrefs

Cf. A030266, A030277, A308031, A308032.

Programs

  • Mathematica
    terms = 31; A[] = 0; Do[A[x] = x + x^2 + x^3 (1 + A[A[x]]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
Nest[x + x^2 + x^3 + x^3 (# /. x -> #) &, O[x], 20][[3]] (* Vladimir Reshetnikov, Aug 08 2019 *)

A308031 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 * (1 + A(A(x))).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 8, 16, 34, 78, 195, 523, 1472, 4284, 12832, 39608, 126406, 418276, 1436230, 5110170, 18785417, 71109917, 276404921, 1101234823, 4493335194, 18773200580, 80320474041, 351906635253, 1578344960050, 7241981076424, 33961826526634, 162615016927284
Offset: 1

Views

Author

Ilya Gutkovskiy, May 10 2019

Keywords

Comments

Shifts left 4 places under COMPOSE transform.

Crossrefs

Cf. A030266, A030277, A308030, A308032.

Programs

  • Mathematica
    terms = 33; A[] = 0; Do[A[x] = x + x^2 + x^3 + x^4 (1 + A[A[x]]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]

A308032 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 + x^4 + x^5 * (1 + A(A(x))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 66, 143, 333, 838, 2250, 6320, 18275, 53925, 161957, 495898, 1554159, 5008758, 16662249, 57311722, 203662894, 745602490, 2801326407, 10760995574, 42141768601, 167955587806, 680843078327, 2808084199432, 11793793850210, 50489830489534
Offset: 1

Views

Author

Ilya Gutkovskiy, May 10 2019

Keywords

Comments

Shifts left 5 places under COMPOSE transform.

Crossrefs

Cf. A030266, A030277, A308030, A308031.

Programs

  • Mathematica
    terms = 35; A[] = 0; Do[A[x] = x + x^2 + x^3 + x^4 + x^5 (1 + A[A[x]]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]

A378575 G.f. satisfies A(x) = x + x*A(A(A(A(A(x))))), so that this sequence shifts left under the 5th self-COMPOSE.

Original entry on oeis.org

1, 1, 5, 45, 545, 7945, 132005, 2423501, 48224129, 1026722489, 23177970949, 551133715197, 13734995332769, 357361170997321, 9677345660994725, 272075021315860781, 7925076713952829697, 238747406787319312025, 7427421640015549840133, 238301672444134819413533, 7875799810817511976148129
Offset: 1

Views

Author

Paul D. Hanna, Dec 01 2024

Keywords

Comments

Conjecture: a(n) == 1 (mod 4) for n >= 1.

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 45*x^4 + 545*x^5 + 7945*x^6 + 132005*x^7 + 2423501*x^8 + 48224129*x^9 + 1026722489*x^10 + ...
where A(x) = x + x*A^5(x).
RELATED SERIES.
A^2(x) = A(A(x)) = x + 2*x^2 + 12*x^3 + 116*x^4 + 1460*x^5 + 21820*x^6 + 369140*x^7 + 6873732*x^8 + 138390908*x^9 + 2976373452*x^10 + ...
A^3(x) = A(A(A(x))) = x + 3*x^2 + 21*x^3 + 219*x^4 + 2885*x^5 + 44483*x^6 + 770269*x^7 + 14610939*x^8 + 298729077*x^9 + 6510526915*x^10 + ...
A^4(x) = A(A(A(A(x)))) = x + 4*x^2 + 32*x^3 + 360*x^4 + 4984*x^5 + 79648*x^6 + 1417768*x^7 + 27500512*x^8 + 572918728*x^9 + 12690763632*x^10 + ...
A^5(x) = A(A(A(A(A(x))))) = x + 5*x^2 + 45*x^3 + 545*x^4 + 7945*x^5 + 132005*x^6 + 2423501*x^7 + 48224129*x^8 + 1026722489*x^9 + ...
...
By formula (4),
A(x) = x + x*A^4(x) + x*A^4(x)*A^8(x) + x*A^4(x)*A^8(x)*A^12(x) + x*A^4(x)*A^8(x)*A^12(x)*A^16(x) + ...
Examples of formula (5), A^n(x) = A^(n+1)(x)/(1 + A^(n+5)(x)):
n=0: x = A(x)/(1 + A(A(A(A(A(x)))))),
n=1: A(x) = A(A(x))/(1 + A(A(A(A(A(A(x))))))),
n=2: A(A(x)) = A(A(A(x)))/(1 + 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(x))))))))),
...
Examples of formula (6), A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+5)(x)):
n=1: A(x) = x*(1 + A(A(A(A(A(x)))))),
n=2: A(A(x)) = x*(1 + A(A(A(A(A(x))))))*(1 + A(A(A(A(A(A(x))))))),
n=3: A(A(A(x))) = x*(1 + A(A(A(A(A(x))))))*(1 + A(A(A(A(A(A(x)))))))*(1 + A(A(A(A(A(A(A(x)))))))),
...

Links

Crossrefs

Cf. A030266, A091713, A196523.

Programs

  • PARI
    /* By definition, A(x) = x + x*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(5, 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^5(x).
(2) A(x) = A(A(x))/(1 + A^6(x)).
(3) A(x) = Series_Reversion( x/(1 + A^4(x)) ).
(4) A(x) = Sum_{n>=0} Product_{k=0..n} A^(4*k)(x).
(5) A^n(x) = A^(n+1)(x) / (1 + A^(n+5)(x)) for n >= 0.
(6) A^n(x) = x*Product_{k>=0..n-1} (1 + A^(k+5)(x)) for n >= 1.

A381666 The generating function A(x) satisfies the functional equation: A(x)+x = x*A(A(x)).

Original entry on oeis.org

0, -1, 1, 0, -2, 1, 10, -13, -70, 163, 585, -2162, -5361, 30588, 49870, -459125, -411370, 7257651, 1513653, -119997558, 56857538, 2062729507, -2444340720, -36662245639, 71849171621, 670108236318, -1904023701457, -12520858710212, 48731008916451, 237412587011506, -1237341547854760
Offset: 0

Views

Author

Thomas Scheuerle, Mar 03 2025

Keywords

Comments

Shifts left under COMPOSE transform with itself.

Examples

			G.f.: A(x) = -x + x^2 - 2*x^4 + x^5 + 10*x^6 + ...
A(A(x)) = x - 2*x^3 + x^4 + 10*x^5 - 13*x^6 + ...

Links

Crossrefs

Cf. A030266 ( A(x)-x = x*A(A(x)) ).
Cf. A347080 ( A(x)-x = x*A(A(-x)) ).
Cf. A381669, A381670.

Programs

  • PARI
    a(n) = { my(A=-1+x); for(i=0, n, A=-1+x*A*subst(A, x, x*A+x*O(x^n))); if(n==0,0,polcoeff(A, n-1))}

Formula

Let a(n) = b(n, 1), with b(1, m) = -1 and b(0, m) = 0, then
b(n, m) = Sum_{k=0..n-1} (-1)^(n-1)*m*binomial(n + m - 1, k)/(n + m - 1) * b(n - k, k).

A381669 The function A(x) = x+(1/2)*x^2-(1/16)*x^4... = Sum_{k >= 0} x^k*a(k)/A381670(k) satisfies the functional equation: x*(A(x)+1) = A(A(x)).

Original entry on oeis.org

0, 1, 1, 0, -1, 1, -1, -1, 113, -19, -1049, 849, 10171, -67975, 183735, 143679, -81627111, -135422127, 3045667427, 341639611, -225862086367, 212228801943, 8911194501081, -5123304557653, -1496818714531027, 6387545555294289, 64005829810291411, -250179519280324047
Offset: 0

Views

Author

Thomas Scheuerle, Mar 03 2025

Keywords

Crossrefs

Cf. A381670 ( denominators ).
Cf. A381666 ( A(x)+x = x*A(A(x)) ).
Cf. A030266 ( A(x)-x = x*A(A(x)) ).
Cf. A347080 ( A(x)-x = x*A(A(-x)) ).

Programs

  • PARI
    compose(v) = polcoeff(subst(Polrev(v),x,Polrev(v)),#v-1)
optimize(v) = { my(r=1,z = v[#v],t = compose(concat(v,r))); while(t<>z, r = r+(z-t)/2; t = compose(concat(v,r)));concat(v,r) }
listA(max_n) = { my(v=[0, 1], out=[0, 1]); while(#v

A381670 The function A(x) = x+(1/2)*x^2-(1/16)*x^4... = Sum_{k >= 0} x^k*A381669(k)/a(k) satisfies the functional equation: x*(A(x)+1) = A(A(x)).

Original entry on oeis.org

1, 1, 2, 1, 16, 16, 64, 16, 1024, 1024, 4096, 2048, 32768, 32768, 131072, 16384, 4194304, 4194304, 16777216, 8388608, 134217728, 134217728, 536870912, 134217728, 8589934592, 8589934592, 34359738368, 17179869184, 274877906944, 274877906944, 1099511627776
Offset: 0

Views

Author

Thomas Scheuerle, Mar 03 2025

Keywords

Comments

Conjecture: All terms are powers of two.

Crossrefs

Cf. A381669 ( numerator ).
Cf. A381666 ( A(x)+x = x*A(A(x)) ).
Cf. A030266 ( A(x)-x = x*A(A(x)) ).
Cf. A347080 ( A(x)-x = x*A(A(-x)) ).

Programs

  • PARI
    compose(v) = polcoeff(subst(Polrev(v),x,Polrev(v)),#v-1)
optimize(v) = { my(r=1,z = v[#v],t = compose(concat(v,r))); while(t<>z, r = r+(z-t)/2; t = compose(concat(v,r)));concat(v,r) }
listA(max_n) = { my(v=[0, 1], out=[1,1]); while(#v

A107097 G.f. satisfies: A(A(x)) = A(x)/(1-x), so that the self-COMPOSE transform generates partial sums (A107098).

Original entry on oeis.org

1, 1, 0, 1, -3, 13, -63, 339, -1982, 12429, -82827, 582589, -4303016, 33240205, -267697961, 2241725581, -19477340744, 175259713769, -1630583565434, 15663877511863, -155168272246709, 1583282220672515, -16623104947488348, 179409709469784087, -1988706708427161585
Offset: 1

Views

Author

Paul D. Hanna, May 12 2005, Jul 23 2011

Keywords

Examples

			G.f.: A(x) = x + x^2 + x^4 - 3*x^5 + 13*x^6 - 63*x^7 + 339*x^8 -+...
If G(x) = series reversion of g.f. A(x) so that A(G(x)) = x, then G(x) begins:
G(x) = x - x^2 + 2*x^3 - 6*x^4 + 23*x^5 - 104*x^6 + 531*x^7 - 2982*x^8 -+...
Compare the functional inverse, G(x), to the arithmetic inverse x/A(x):
x/A(x) = 1 - x + x^2 - 2*x^3 + 6*x^4 - 23*x^5 + 104*x^6 - 531*x^7 + 2982*x^8 -+...

Crossrefs

Cf. A107098.

Programs

  • PARI
    {a(n)=local(A,B,F);if(n<1,0,F=x+2*x^2-3*x^3+x*O(x^n);A=F; for(j=0,n, for(i=0,j,B=serreverse(A);A=(A+subst(B,x, A/(1-x)))/2); A=round(A));polcoeff(A,n,x))}
  • PARI
    /* A(x) = x + A(x)*Series_Reversion(A(x)): */
{a(n)=local(A=x+x^2);for(i=1,n,A=x+A*serreverse(A+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = x + A(x)*Series_Reversion(A(x)).
Given g.f. A(x), let G(x) = Series_Reversion(A(x)), then G(x) satisfies:
(1) G(x) = 1 - x/A(x),
(2) G(x) = x - x*G(G(x)),
(3) -G(-x) is the g.f. of A030266, which shifts left under self-COMPOSE.

Extensions

Initial zero removed and offset changed to 1 by Paul D. Hanna, Jul 23 2011

A107098 The self-COMPOSE transform of A107097 and also the partial sums of A107097: g.f. A(x) = G(G(x)) = G(x)/(1-x) where G(x) is the g.f. of A107097.

Original entry on oeis.org

0, 1, 2, 2, 3, 0, 13, -50, 289, -1693, 10736, -72091, 510498, -3792518, 29447687, -238250274, 2003475307, -17473865437, 157785848332, -1472797717102, 14191079794761, -140977192451948, 1442305028220567, -15180799919267781, 164228909550516306, -1824477798876645279
Offset: 0

Views

Author

Paul D. Hanna, May 12 2005

Keywords

Examples

			Series reversion of g.f.:
x + 2*x^2 + 2*x^3 + 3*x^4 + 13*x^6 - 50*x^7 + 289*x^8 -+...
equals (G(-x)+x)/x where G(x) is g.f. for A030266:
x - 2*x^2 + 6*x^3 - 23*x^4 + 104*x^5 - 531*x^6 +-...

Crossrefs

Cf. A107097, A030266.

Programs

  • PARI
    {a(n)=local(A,B,F);if(n<1,0,F=x+2*x^2-3*x^3+x*O(x^n);A=F; for(j=0,n, for(i=0,j,B=serreverse(A);A=(A+subst(B,x,A/(1-x)))/2); A=round(A));polcoeff(A/(1-x),n,x))}

Formula

G.f. A(x) = series-reversion of (G(-x)+x)/x where G(x) is g.f. for A030266.

A242794 a(n) = [x^n] ( 1 + x*A(x)^n )^(n+1) / (n+1) for n>=0, with a(0)=1.

Original entry on oeis.org

1, 1, 3, 22, 257, 3986, 75304, 1653086, 40979297, 1126004203, 33856704386, 1103686134563, 38734891315775, 1455569736467094, 58304721086789654, 2480233978808257526, 111686585878084164913, 5308774844414927594856, 265682854185812938555354, 13966882165871163036529423
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 + 3*x^2 + 22*x^3 + 257*x^4 + 3986*x^5 + 75304*x^6 +...
Form a table of coefficients of x^k in (1 + x*A(x)^n)^(n+1) like so:
n=0: [1, 1,   0,    0,     0,      0,       0,        0, ...];
n=1: [1, 2,   3,    8,    51,    564,    8539,   159226, ...];
n=2: [1, 3,   9,   34,   210,   2118,   30245,   544962, ...];
n=3: [1, 4,  18,   88,   575,   5472,   73242,  1263604, ...];
n=4: [1, 5,  30,  180,  1285,  12016,  151820,  2490390, ...];
n=5: [1, 6,  45,  320,  2520,  23916,  290162,  4518600, ...];
n=6: [1, 7,  63,  518,  4501,  44310,  527128,  7834548, ...];
n=7: [1, 8,  84,  784,  7490,  77504,  922096, 13224688, ...];
n=8: [1, 9, 108, 1128, 11790, 129168, 1561860, 21921156, ...]; ...
then this sequence is formed from the main diagonal:
[1/1, 2/2, 9/3, 88/4, 1285/5, 23916/6, 527128/7, 13224688/8, ...].

Crossrefs

Cf. A242795.

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)^(m+1))[m+1]/(m+1));A[n+1]}
for(n=0,25,print1(a(n),", "))
Previous Showing 31-40 of 44 results. Next

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