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 11-12 of 12 results.

A277310 G.f. satisfies: A(x - 4*A(x)^2) = x - 3*A(x)^2.

Original entry on oeis.org

1, 1, 10, 141, 2422, 47562, 1031764, 24214405, 606444990, 16055089470, 446238074892, 12955112773554, 391332183548956, 12261884937532340, 397576302315045800, 13313017677172350965, 459635990935574444942, 16339309997761322057206, 597340515437542895494748, 22435278085988347895795526, 864900964565994975048855444, 34195693888939483596581262668, 1385553440866978431053220575128
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 10*x^3 + 141*x^4 + 2422*x^5 + 47562*x^6 + 1031764*x^7 + 24214405*x^8 + 606444990*x^9 + 16055089470*x^10 +...
such that A(x - 4*A(x)^2) = x - 3*A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 21*x^4 + 302*x^5 + 5226*x^6 + 102788*x^7 + 2226973*x^8 + 52126582*x^9 + 1301232638*x^10 + 34328704796*x^11 + 950803699394*x^12 + 27510261070028*x^13 + 828332416917876*x^14 + 25876801064095496*x^15 + 836682915170627501*x^16 +...
A(x - 4*A(x)^2) = x - 3*x^2 - 6*x^3 - 63*x^4 - 906*x^5 - 15678*x^6 - 308364*x^7 - 6680919*x^8 - 156379746*x^9 - 3903697914*x^10 +...
which equals x - 3*A(x)^2.
Series_Reversion(x - 4*A(x)^2) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9688*x^5 + 190248*x^6 + 4127056*x^7 + 96857620*x^8 + 2425779960*x^9 + 64220357880*x^10 +...
which equals -3*x + 4*A(x).
A( 4*A(x) - 3*x ) = x + 5*x^2 + 58*x^3 + 921*x^4 + 17494*x^5 + 374994*x^6 + 8793460*x^7 + 221393569*x^8 + 5912166718*x^9 + 166058455158*x^10 + 4876311925036*x^11 + 149037482367530*x^12 + 4724877954111836*x^13 + 154959634972646340*x^14 + 5246331138228520168*x^15 +...
which equals  sqrt( A(x) - x ).

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-4*F^2) + 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 4*A(x) - 3*x )^2.
(2) A(x) = 3*x/4 + 1/4 * Series_Reversion(x - 4*A(x)^2).
(3) R(x) = 4*x/3 - 1/3 * Series_Reversion(x - 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x - R(x) ) ) = 4*x - 3*R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 4^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 4^k.

A277311 G.f. satisfies: A(x - 5*A(x)^2) = x - 4*A(x)^2.

Original entry on oeis.org

1, 1, 12, 200, 4034, 92752, 2353272, 64579809, 1891598860, 58591554652, 1906271367296, 64816527248936, 2294331974613872, 84290267670946720, 3206227129084419920, 126022120854865417140, 5110001578581607976400, 213458728365617240931360, 9175021814527973211291880, 405366362599820848509766760, 18392202994173383123235536800, 856255190568423353781484124240
Offset: 1

Views

Author

Paul D. Hanna, Oct 12 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 12*x^3 + 200*x^4 + 4034*x^5 + 92752*x^6 + 2353272*x^7 + 64579809*x^8 + 1891598860*x^9 + 58591554652*x^10 +...
such that  A(x - 5*A(x)^2) = x - 4*A(x)^2.
A(x)^2 = x^2 + 2*x^3 + 25*x^4 + 424*x^5 + 8612*x^6 + 198372*x^7 + 5028864*x^8 + 137705810*x^9 + 4022209822*x^10 + 124205854376*x^11 + 4028545272136*x^12 + 136566005356212*x^13 + 4820263259998720*x^14 + 176614868022441920*x^15 +...
A(x - 5*A(x)^2) = x - 4*x^2 - 8*x^3 - 100*x^4 - 1696*x^5 - 34448*x^6 - 793488*x^7 - 20115456*x^8 - 550823240*x^9 - 16088839288*x^10 +...
which equals x - 4*A(x)^2.
Series_Reversion(x - 5*A(x)^2) = x + 5*x^2 + 60*x^3 + 1000*x^4 + 20170*x^5 + 463760*x^6 + 11766360*x^7 + 322899045*x^8 + 9457994300*x^9 +...
which equals  5*A(x) - 4*x.
A( 5*A(x) - 4*x ) = x + 6*x^2 + 82*x^3 + 1525*x^4 + 33864*x^5 + 848402*x^6 + 23259832*x^7 + 685028874*x^8 + 21411099560*x^9 + 704295189492*x^10 +24234549363096*x^11 + 868423052983416*x^12 + 32296557071230392*x^13 + 1243216715481216720*x^14 + 49428242214109804120*x^15 +...
which equals  sqrt( A(x) -x ).

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309, A277310.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-5*F^2) + 4*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + A( 5*A(x) - 4*x )^2.
(2) A(x) = 4*x/5 + 1/5 * Series_Reversion(x - 5*A(x)^2).
(3) R(x) = 5*x/4 - 1/4 * Series_Reversion(x - 4*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x - R(x) ) ) = 5*x - 4*R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 5^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^k.
Previous Showing 11-12 of 12 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.