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.

A275755 G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^5 ).

Original entry on oeis.org

1, 1, 2, 6, 19, 65, 234, 873, 3346, 13099, 52154, 210541, 859768, 3545263, 14741148, 61736903, 260192880, 1102704585, 4696416190, 20090502706, 86285786519, 371917832707, 1608317086940, 6975728777332, 30338392601498, 132277349730004, 578075052215714, 2531710609461484, 11109852467209553, 48843541287179595, 215108137824940916, 948874606956945665, 4191979050580762418, 18545890698661636784, 82159569800859439840, 364432560308538162214, 1618431087549954575022
Offset: 1

Views

Author

Paul D. Hanna, Aug 20 2016

Keywords

Comments

Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 65*x^6 + 234*x^7 + 873*x^8 + 3346*x^9 + 13099*x^10 + 52154*x^11 + 210541*x^12 + 859768*x^13 + 3545263*x^14 +...
such that A(x) = x + A( A(x)^2 - A(x)^5 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 54*x^6 + 192*x^7 + 710*x^8 + 2702*x^9 + 10515*x^10 + 41660*x^11 + 167483*x^12 + 681532*x^13 + 2801816*x^14 +...
A(x)^5 = x^5 + 5*x^6 + 20*x^7 + 80*x^8 + 320*x^9 + 1286*x^10 + 5210*x^11 + 21285*x^12 + 87655*x^13 + 363660*x^14 + 1518952*x^15 +...
A(x^2 - x^5) = x^2 + x^4 - x^5 + 2*x^6 - 2*x^7 + 6*x^8 - 6*x^9 + 20*x^10 - 24*x^11 + 71*x^12 - 95*x^13 + 270*x^14 - 392*x^15 + 1063*x^16 - 1662*x^17 +...
where Series_Reversion(A(x)) = x - A(x^2 - x^5).

Links

Crossrefs

Cf. A275756, A275757, A275758, A141200.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = x + subst(A,x, A^2 - A^5 +x*O(x^n))); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f. satisfies:
(1) A(x - A(x^2 - x^5)) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2-x^5)^(n+1) / (n+1)!.
(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^2-x^5)^(n+1)/x / (n+1)! ).

A275756 G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^6 ).

Original entry on oeis.org

1, 1, 2, 6, 20, 71, 264, 1018, 4032, 16305, 67042, 279444, 1178088, 5014596, 21521488, 93027025, 404630318, 1769704106, 7778030834, 34335337802, 152168657438, 676796514510, 3019945599904, 13515300673984, 60649985907334, 272847379282493, 1230295797205452, 5559373120441048, 25171114275512520, 114177375142080814, 518806321789317040, 2361183952087172306, 10762422470020855820, 49125407360603361370, 224533932290057629076, 1027553322543206612019, 4708070541211739962738, 21595828228486254332762
Offset: 1

Views

Author

Paul D. Hanna, Aug 20 2016

Keywords

Comments

Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
Compare to: C(x) = x + C( C(x)^3 - C(x)^9 ) holds when C(x) = x + C(x)^3 is a g.f. of the ternary tree numbers (A001764).

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 71*x^6 + 264*x^7 + 1018*x^8 + 4032*x^9 + 16305*x^10 + 67042*x^11 + 279444*x^12 + 1178088*x^13 + 5014596*x^14 +...
such that A(x) = x + A( A(x)^2 - A(x)^6 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 206*x^7 + 786*x^8 + 3088*x^9 + 12408*x^10 + 50754*x^11 + 210639*x^12 + 884784*x^13 + 3754424*x^14 +...
A(x)^6 = x^6 + 6*x^7 + 27*x^8 + 116*x^9 + 495*x^10 + 2112*x^11 + 9035*x^12 + 38820*x^13 + 167628*x^14 + 727480*x^15 + 3172455*x^16 +...
A(x^2 - x^6) = x^2 + x^4 + x^6 + 4*x^8 + 14*x^10 + 48*x^12 + 170*x^14 + 628*x^16 + 2382*x^18 + 9202*x^20 + 36098*x^22 + 143484*x^24 + 576638*x^26 + 2339050*x^28 +...
where Series_Reversion(A(x)) = x - A(x^2 - x^6).

Links

Crossrefs

Cf. A275755, A275757, A275758, A141200.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = x + subst(A,x, A^2 - A^6 +x*O(x^n))); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f. satisfies:
(1) A(x - A(x^2 - x^6)) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2-x^6)^(n+1) / (n+1)!.
(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^2-x^6)^(n+1)/x / (n+1)! ).

A275758 G.f. satisfies: A(x) = x + A( A(x)^4 - A(x)^10 ).

Original entry on oeis.org

1, 1, 4, 21, 126, 817, 5574, 39418, 286286, 2122491, 15995696, 122166551, 943430560, 7353998931, 57783603764, 457176705018, 3639000808140, 29119701312548, 234120338807316, 1890257713736568, 15319612051101438, 124583720191974904, 1016307862050772614, 8314217332992596050, 68193993494598345010, 560671685990956975367, 4619857060146629819160, 38144728242794104501561, 315546193363448088862064, 2614910268303053285326541
Offset: 1

Views

Author

Paul D. Hanna, Aug 20 2016

Keywords

Comments

Compare to: G(x) = x + G( G(x)^4 - G(x)^16 ) holds when G(x) = x + G(x)^4 is a g.f. of A002293.
Compare to: F(x) = x + F( F(x)^3 - F(x)^9 ) holds when F(x) = x + F(x)^3 is a g.f. of the ternary tree numbers (A001764).

Examples

			G.f.: A(x) = x + x^4 + 4*x^7 + 21*x^10 + 126*x^13 + 817*x^16 + 5574*x^19 + 39418*x^22 + 286286*x^25 + 2122491*x^28 + 15995696*x^31 + 122166551*x^34 +...
such that A(x) = x + A( A(x)^4 - A(x)^10 ).
RELATED SERIES.
A(x)^4 = x^4 + 4*x^7 + 22*x^10 + 136*x^13 + 901*x^16 + 6248*x^19 + 44758*x^22 + 328520*x^25 + 2457286*x^28 + 18659736*x^31 + 143455026*x^34 +...
A(x)^10 = x^10 + 10*x^13 + 85*x^16 + 690*x^19 + 5520*x^22 + 44002*x^25 + 351045*x^28 + 2808040*x^31 + 22537355*x^34 + 181530280*x^37 + 1467320874*x^40 +...
A(x^4 - x^10) = x^4 - x^10 + x^16 - 4*x^22 + 10*x^28 - 32*x^34 + 106*x^40 - 350*x^46 + 1211*x^52 - 4242*x^58 + 15083*x^64 - 54404*x^70 + 198114*x^76 +...
where Series_Reversion(A(x)) = x - A(x^4 - x^10).

Links

Crossrefs

Cf. A275755, A275756, A275757.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, 3*n, A = x + subst(A, x, A^4 - A^10 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x - A(x^4 - x^10)) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^4-x^10)^(n+1) / (n+1)!.
(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^4-x^10)^(n+1)/x / (n+1)! ).
Showing 1-3 of 3 results.

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