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 1-8 of 8 results.

A271930 G.f. A(x) satisfies: A(x) = A( x^2 + 6*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 3, 15, 90, 597, 4221, 31185, 237897, 1859568, 14816637, 119892942, 982565883, 8138777166, 68028775587, 573078135996, 4860507197700, 41470162208814, 355695498901179, 3065210379987489, 26525947283576640, 230425563258798840, 2008561878414115803, 17563090615911038115, 154014411705019299450, 1354142406561753259035, 11934928413519024726252, 105426063390991627937457, 933206579920813459523994, 8276480132736299734057275, 73535083052134446419214960
Offset: 1

Views

Author

Paul D. Hanna, Apr 16 2016

Keywords

Comments

Compare the g.f. to the following related identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
(2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).

Examples

			G..f.: A(x) = x + 3*x^2 + 15*x^3 + 90*x^4 + 597*x^5 + 4221*x^6 + 31185*x^7 + 237897*x^8 + 1859568*x^9 + 14816637*x^10 + 119892942*x^11 + 982565883*x^12 +...
where A(x)^2 = A( x^2 + 6*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 39*x^4 + 270*x^5 + 1959*x^6 + 14724*x^7 + 113706*x^8 + 896994*x^9 + 7198257*x^10 + 58580766*x^11 + 482345937*x^12 + 4011023556*x^13 + 33637887441*x^14 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 3*x^2 + 3*x^3 - 3*x^5 + 9*x^7 - 33*x^9 + 126*x^11 - 513*x^13 + 2214*x^15 - 9876*x^17 + 45045*x^19 - 209493*x^21 +...+ A264412(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 3*x^2 where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.

Links

Crossrefs

Cf. A264412, A271935, A271957, A271931, A271934.

Programs

  • PARI
    {a(n) = my(A=x,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 6*X*A^2)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A( x*G(x^2) - 3*x^2 ) = x, where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.
a(n) ~ c * d^n / n^(3/2), where d = 9.35010183959428615991060685319... and c = 0.0902227396498060205291555743... . - Vaclav Kotesovec, Apr 18 2016

A271932 G.f. A(x) satisfies: A(x) = A( x^7 + 7*x*A(x)^7 )^(1/7), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 4, 20, 110, 638, 3828, 23515, 146970, 930820, 5957325, 38452405, 249944939, 1634287025, 10739831400, 70884562683, 469622328252, 3121694320866, 20811920304961, 139115729296575, 932107335003790, 6258662787526655, 42105353650697301, 283765005631661148, 1915495724241980280, 12949332513585521217, 87661142189041380207, 594176943178375193748, 4032121696383579351905, 27392082325012470506385, 186276500908841717917320
Offset: 1

Views

Author

Paul D. Hanna, Apr 16 2016

Keywords

Comments

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + p*x*G(x)^p )^(1/p) with G(0)=0, G'(0)=1 (conjecture).

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 20*x^4 + 110*x^5 + 638*x^6 + 3828*x^7 + 23515*x^8 + 146970*x^9 + 930820*x^10 + 5957325*x^11 + 38452405*x^12 +...
where A(x)^7 = A( x^7 + 7*x*A(x)^7 ).
RELATED SERIES.
A(x)^7 = x^7 + 7*x^8 + 49*x^9 + 343*x^10 + 2401*x^11 + 16807*x^12 + 117649*x^13 + 823544*x^14 + 5764822*x^15 + 40353901*x^16 + 282478679*x^17 + 1977362758*x^18 + 13841640148*x^19 + 96892304579*x^20 + 678252720401*x^21 +...

Links

Crossrefs

Cf. A271931, A271933.

Programs

  • PARI
    {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^7 + 7*X*A^7)^(1/7) ); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

A271933 G.f. A(x) satisfies: A(x) = A( x^11 + 11*x*A(x)^11 )^(1/11), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 6, 46, 391, 3519, 32844, 314364, 3065049, 30309929, 303099290, 3058547381, 31095231708, 318128139796, 3272175152355, 33812476576290, 350804444501589, 3652493334187197, 38148263715573364, 399552867370295155, 4195305107766973240, 44150591852677070280, 465588059585378099226, 4919039064854516328821, 52059830109088065802395, 551834199223958450647359, 5857932269440676202573084
Offset: 1

Views

Author

Paul D. Hanna, Apr 16 2016

Keywords

Comments

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + p*x*G(x)^p )^(1/p) with G(0)=0, G'(0)=1 (conjecture).

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 46*x^4 + 391*x^5 + 3519*x^6 + 32844*x^7 + 314364*x^8 + 3065049*x^9 + 30309929*x^10 + 303099290*x^11 + 3058547381*x^12 +...
where A(x)^11 = A( x^11 + 11*x*A(x)^11 ).
RELATED SERIES.
A(x)^11 = x^11 + 11*x^12 + 121*x^13 + 1331*x^14 + 14641*x^15 + 161051*x^16 + 1771561*x^17 + 19487171*x^18 + 214358881*x^19 + 2357947691*x^20 + 25937424601*x^21 + 285311670612*x^22 + 3138428376754*x^23 + 34522712144657*x^24 +...

Links

Crossrefs

Cf. A271931, A271932.

Programs

  • PARI
    {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^11 + 11*X*A^11)^(1/11) ); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

A271934 G.f. A(x) satisfies: A(x) = A( x^3 + 6*x*A(x)^3 )^(1/3), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 2, 8, 38, 196, 1064, 5988, 34632, 204672, 1231082, 7514052, 46433088, 289976404, 1827459072, 11608240000, 74249294704, 477826080368, 3091718252320, 20101537759256, 131262924427560, 860504352317040, 5661120688863216, 37363827222888640, 247331149667685440, 1641642515512685408, 10923380539408947456, 72850297774044995328, 486886413558080754198, 3260469757311730139044, 21874082006618739609864
Offset: 1

Views

Author

Paul D. Hanna, Apr 16 2016

Keywords

Comments

Compare the g.f. to the following identity:
C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + 2*x^2 + 8*x^3 + 38*x^4 + 196*x^5 + 1064*x^6 + 5988*x^7 + 34632*x^8 + 204672*x^9 + 1231082*x^10 + 7514052*x^11 + 46433088*x^12 +...
where A(x)^3 = A( x^3 + 6*x*A(x)^3 ).
RELATED SERIES.
A(x)^3 = x^3 + 6*x^4 + 36*x^5 + 218*x^6 + 1332*x^7 + 8208*x^8 + 50984*x^9 + 319056*x^10 + 2010528*x^11 + 12750950*x^12 + 81348948*x^13 + 521839944*x^14 + 3364421812*x^15 + 21791976192*x^16 +...
Series reversion of A(x) equals B(x) - 2*x^3/B(x) where
B(x) = x + 2*x^4 - 20*x^7 + 302*x^10 - 5436*x^13 + 108072*x^16 - 2286160*x^19 + 50475256*x^22 - 1149822240*x^25 + 26825146770*x^28 - 637700980612*x^31 + 15391872726072*x^34 - 376193675011356*x^37 + 9291840570002312*x^40 +...

Crossrefs

Cf. A271931.

Programs

  • PARI
    {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^3 + 6*X*A^3)^(1/3) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A( B(x) - 2*x^3/B(x) ) = x, where B(x) is described in the examples.

A271935 G.f. A(x) satisfies: A(x) = A( x^2 + 8*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 4, 26, 200, 1691, 15204, 142710, 1382568, 13721765, 138802136, 1425785270, 14832383488, 155947271878, 1654494195340, 17690004381000, 190426309700616, 2062071992480208, 22447191471665160, 245501068961175090, 2696300196714320520, 29725402250477117175, 328835072363241763920
Offset: 1

Views

Author

Paul D. Hanna, Apr 16 2016

Keywords

Comments

Compare the g.f. to the following related identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
(2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).

Examples

			G..f.: A(x) = x + 4*x^2 + 26*x^3 + 200*x^4 + 1691*x^5 + 15204*x^6 + 142710*x^7 + 1382568*x^8 + 13721765*x^9 + 138802136*x^10 + 1425785270*x^11 + ...
where A(x)^2 = A( x^2 + 8*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 8*x^3 + 68*x^4 + 608*x^5 + 5658*x^6 + 54336*x^7 + 534984*x^8 + 5373824*x^9 + 54866075*x^10 + 567775856*x^11 + 5942353444*x^12 + ...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 4*x^2 + 6*x^3 - 15*x^5 + 90*x^7 - 660*x^9 + 5310*x^11 - 45765*x^13 + 413640*x^15 - 3864345*x^17 + 37014120*x^19 + ... + A264413(n)*x^(2*n+1) + ...
such that B(x) = x*G(x^2) - 4*x^2 where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.
From _Paul D. Hanna_, May 20 2024: (Start)
The series (A(x)/x)^(1/4) seems to consist solely of integer coefficients
(A(x)/x)^(1/4) = 1 + x + 5*x^2 + 34*x^3 + 268*x^4 + 2305*x^5 + 20988*x^6 + 198891*x^7 + 1941111*x^8 + 19377707*x^9 + 196936775*x^10 + ...
and continues to be integral for at least the initial 400 coefficients. (End)

Links

Crossrefs

Cf. A264413, A271930, A271957, A271931, A271934.

Programs

  • PARI
    {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 8*X*A^2)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A( x*G(x^2) - 4*x^2 ) = x, where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.

A271957 G.f. A(x) satisfies: A(x) = A( x^2 + 10*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 5, 40, 375, 3845, 41825, 474450, 5552250, 66548785, 812875800, 10082125950, 126637168125, 1607562407775, 20591392666250, 265810034489750, 3454516382881875, 45162288467005155, 593528625987396725, 7836767285955169200, 103908861022437312375, 1382961699685548183750, 18469547560714428659250, 247433242662040209056250, 3324296142183357299203125, 44779542961314348791789400, 604655933814703316140014375
Offset: 1

Views

Author

Paul D. Hanna, Apr 17 2016

Keywords

Comments

Compare the g.f. to the following related identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
(2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).

Examples

			G..f.: A(x) = x + 5*x^2 + 40*x^3 + 375*x^4 + 3845*x^5 + 41825*x^6 + 474450*x^7 + 5552250*x^8 + 66548785*x^9 + 812875800*x^10 + 10082125950*x^11 + 126637168125*x^12 +...
where A(x)^2 = A( x^2 + 10*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 10*x^3 + 105*x^4 + 1150*x^5 + 13040*x^6 + 152100*x^7 + 1815375*x^8 + 22078750*x^9 + 272728845*x^10 + 3412891200*x^11 + 43178951325*x^12 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 5*x^2 + 10*x^3 - 45*x^5 + 450*x^7 - 5535*x^9 + 75600*x^11 - 1106100*x^13 + 16953750*x^15 +...+ A264414(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 5*x^2 where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.

Links

Crossrefs

Cf. A264414, A271930, A271935, A271931, A271934.

Programs

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

Formula

G.f. A(x) satisfies: A( x*G(x^2) - 5*x^2 ) = x, where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.

A370545 Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).

Original entry on oeis.org

1, 1, 4, 21, 125, 801, 5388, 37518, 268109, 1955000, 14487754, 108794169, 826054062, 6331064385, 48914088750, 380555960864, 2978892961194, 23444095375593, 185394136871818, 1472396312841250, 11739089289817538, 93921736129064325, 753845680317416682, 6068255413854119432
Offset: 1

Views

Author

Paul D. Hanna, Mar 26 2024

Keywords

Comments

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 125*x^5 + 801*x^6 + 5388*x^7 + 37518*x^8 + 268109*x^9 + 1955000*x^10 + 14487754*x^11 + 108794169*x^12 + ...
where A(x)^5 = A( x^5 + 5*A(x)^6 ).
RELATED SERIES.
A(x)^5 = x^5 + 5*x^6 + 30*x^7 + 195*x^8 + 1330*x^9 + 9376*x^10 + 67720*x^11 + ...
A(x)^6 = x^6 + 6*x^7 + 39*x^8 + 266*x^9 + 1875*x^10 + 13542*x^11 + 99654*x^12 + ...
Let B(x) be the series reversion of A(x), A(B(x)) = x, which begins
B(x) = x - x^2 - 2*x^3 - 6*x^4 - 21*x^5 - 80*x^6 - 320*x^7 - 1326*x^8 - 5637*x^9 - 24434*x^10 - ... + (-1)^(n-1)*A352703(n-1)*x^n + ...
then B(x)^5 + 5*x^6 = B(x^5).
Let C(x) = x^2/B(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 802*x^7 + 3553*x^8 + 15994*x^9 + 72886*x^10 + ... + A091200(n-1)*x^n + ...
where A(x^2/C(x)) = x and C(A(x)) = A(x)^2/x,
then C(x)^5 = C(x^5)/(1 - 5*C(x^5)/x^4).

Links

Crossrefs

Cf. A352703, A091200, A271931, A370546.

Programs

  • PARI
    {a(n) = my(A=x+x^2); for(m=1, n, A=truncate(A); A = subst(A, x, x^5 + 5*A^6 +x^5*O(x^m))^(1/5) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

A356780 Coefficients in the odd function A(x) such that: A(x) = A( x^2 + 2*x^2*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 303, 1223, 5085, 21623, 93585, 410894, 1825682, 8193544, 37087449, 169114547, 776110247, 3581944258, 16614576945, 77410877233, 362126147797, 1700179143293, 8008689767674, 37838553977426, 179268540549758, 851478474635404, 4053760582437106
Offset: 1

Views

Author

Paul D. Hanna, Aug 27 2022

Keywords

Comments

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f. A(x) = x + x^3 + 2*x^5 + 6*x^7 + 21*x^9 + 78*x^11 + 303*x^13 + 1223*x^15 + 5085*x^17 + 21623*x^19 + 93585*x^21 + ...
where A(x)^2 = A( x^2 + 2*x^2*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^4 + 5*x^6 + 16*x^8 + 58*x^10 + 222*x^12 + 882*x^14 + 3616*x^16 + 15205*x^18 + 65220*x^20 + ...
x^2 + 2*x^2*A(x)^2 = x^2 + 2*x^4 + 4*x^6 + 10*x^8 + 32*x^10 + 116*x^12 + 444*x^14 + 1764*x^16 + 7232*x^18 + 30410*x^20 + ...
Let G(x) = Series_Reversion( A(x) ) then
G(x) = x - x^3 + x^5 - 2*x^7 + 4*x^9 - 7*x^11 + 12*x^13 - 23*x^15 + 45*x^17 - 84*x^19 + 157*x^21 - 302*x^23 + 584*x^25 - 1121*x^27 + ...
where G(x)^2 = G(x^2)/(1 + 2*x^2) and G(A(x)) = x.

Crossrefs

Cf. A000108, A356781, A271931, A271932, A271933.

Programs

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

Formula

G.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) satisfies:
(1) A(x) = sqrt( A( x^2 + 2*x^2*A(x)^2 ) ).
(2) G(x) = sqrt( G(x^2) / (1 + 2*x^2) ), where A(G(x)) = x.
Showing 1-8 of 8 results.

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