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.

A375443 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^3 )/(1-2*x).

Original entry on oeis.org

1, 1, 2, 6, 21, 77, 290, 1122, 4462, 18210, 76028, 323524, 1398071, 6115707, 27008516, 120162616, 537702116, 2417043444, 10904533054, 49343555890, 223851302500, 1017798552096, 4637127493554, 21167261603078, 96799606576699, 443460169286639, 2035144213216892, 9355941004378324
Offset: 0

Views

Author

Paul D. Hanna, Aug 18 2024

Keywords

Comments

Compare to M(x)^2 = M( x^2/(1-2*x) )/(1-2*x), where M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).
Compare to C(x)^2 = C( x^2/(1-2*x)^2 )/(1-2*x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + 3*x^2 + 9*x^3 + 31*x^4 + 117*x^5 + 459*x^6 + 1835*x^7 + 7449*x^8 + 30711*x^9 + 128601*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^3 )/(1-2*x).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 58*x^4 + 220*x^5 + 854*x^6 + 3384*x^7 + 13693*x^8 + 56546*x^9 + 237897*x^10 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 117*x^4 + 459*x^5 + 1835*x^6 + 7449*x^7 + 30711*x^8 + ... + A375453(n+1)*x^n + ...
SPECIFIC VALUES.
Given the radius of convergence r = 0.2051227438492708078605991264519...,
A(r) = 1.6956207695598620574163671001175353426181793882085...
  where r = (1-2*r)^3 and A(r) = 1/(1-2*r).
A(1/5) = 1.51884977058839576453094931523796453209831069839...
  where A(1/5)^2 = (5/3)*A(5/27).
A(1/6) = 1.29543251347110009761686143135328534086163706795...
  where A(1/6)^2 = (6/4)*A(6/64).
A(1/7) = 1.22025427535592887335278669533719663766721910803...
  where A(1/7)^2 = (7/5)*A(7/125).
A(1/10) = 1.12934836581956838019397695630366800332615427708...
  where A(1/10)^2 = (10/8)*A(10/512).

Links

Crossrefs

Cf. A001006, A000108, A375453, A375444, A375445.

Programs

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

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2/(1-2*x)^3 )/(1-2*x).
(2) A(x)^4 = A( x^4*y^3 )*y where y = (1-2*x)/((1-4*x)*(1-2*x+2*x^2)).
(3) A(x^2 + 2*x^3) = A( x/(1+2*x) )^2 / (1+2*x).
The radius of convergence r satisfies r = (1 - 2*r)^3, where A(r) = 1/(1-2*r) and r = (1/12)*(6 + (6*sqrt(87) - 54)^(1/3) - (6*sqrt(87) + 54)^(1/3)) = 0.20512274384927080786...

A375454 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^4 ).

Original entry on oeis.org

1, 4, 14, 56, 253, 1188, 5598, 26456, 126278, 611640, 3010948, 15058064, 76399263, 392524428, 2038142346, 10674131464, 56281299098, 298286598920, 1586959692508, 8466559886448, 45260129274602, 242296862848648, 1298487003814300, 6964374684442416, 37378818578434617, 200745991803248388
Offset: 1

Views

Author

Paul D. Hanna, Aug 17 2024

Keywords

Comments

Compare to: F(x)^2 = F( x^2/(1-2*x) ), where F(x) = x*M(x) and M(x) = 1 + x*M(x) + x^2*M(x)^2 is the Motzkin function (A001006).
Compare to: G(x)^2 = G( x^2/(1-2*x)^2 ), where G(x) = x*C(x)^2 and C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

Examples

			G.f.: A(x) = x + 4*x^2 + 14*x^3 + 56*x^4 + 253*x^5 + 1188*x^6 + 5598*x^7 + 26456*x^8 + 126278*x^9 + 611640*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^4 ).
RELATED SERIES.
A(x)^2 = x^2 + 8*x^3 + 44*x^4 + 224*x^5 + 1150*x^6 + 5968*x^7 + 30920*x^8 + 159296*x^9 + 818013*x^10 + ...
(A(x)/x)^(1/2) = 1 + 2*x + 5*x^2 + 18*x^3 + 78*x^4 + 348*x^5 + 1551*x^6 + 6982*x^7 + 32114*x^8 + 151620*x^9 + 734458*x^10 + ...
(A(x)/x)^(1/4) = 1 + x + 2*x^2 + 7*x^3 + 30*x^4 + 130*x^5 + 561*x^6 + 2460*x^7 + 11115*x^8 + 51948*x^9 + 250551*x^10 + ...
x/Series_Reversion( A( x^4/(1-2*x)^4 )^(1/4) ) = 1 + 2*x + x^4 - 2*x^8 + 9*x^12 - 46*x^16 + 251*x^20 - 1467*x^24 + 9001*x^28 - 56961*x^32 + 369035*x^36 + ...
SPECIFIC VALUES.
A(t) = 3/4 at t = 0.1736940609090204697398931237543698538457793088071...
A(t) = 3/5 at t = 0.1683900940132911881249251740473132322447213579710...
A(t) = 1/2 at t = 0.1619792168710406277922262531667140037108384145069...
A(t) = 2/5 at t = 0.1519644942152899698264943158671103683281536948439...
A(t) = 1/4 at t = 0.1256102247935771858127425480259396391143977190820...
A(1/6) = 0.56831064552606196969057398988138945640151282529741...
where A(1/6)^2 = A(9/64).
A(1/7) = 0.33620864108171743638518920200481354359529601205566...
where A(1/7)^2 = A(49/625).
A(1/8) = 0.24749365203461325611367946855098276802746512221191...
where A(1/8)^2 = A(4/81).
A(1/9) = 0.19726025709005021216217281111162473370492543981591...
where A(1/9)^2 = A(81/2401).
A(1/10) = 0.1643908422523431149995416239752018754267879073230...
where A(1/10)^2 = A(25/1024).

Links

Crossrefs

Cf. A001006, A000108, A375453, A375455, A375456.

Programs

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

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2/(1-2*x)^4 ).
(2) A(x)^4 = A( x^4*(1-2*x)^8/((1-2*x)^4 - 2*x^2)^4 ).
(3) A(x^2 + 4*x^3 + 4*x^4) = A( x/(1+2*x) )^2.
The radius of convergence r satisfies r = (1 - 2*r)^4, where A(r) = 1 and r = 0.17610056436947880725475...

A375455 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 ).

Original entry on oeis.org

1, 5, 20, 90, 470, 2566, 13885, 74435, 400530, 2183930, 12112167, 68351005, 392055575, 2281947435, 13450584580, 80110426698, 481032299830, 2905955107950, 17629836770715, 107254895106265, 653597751574541, 3986386422481665, 24321398369358070, 148386468804372420, 905156432977350225
Offset: 1

Views

Author

Paul D. Hanna, Aug 17 2024

Keywords

Comments

Compare to: F(x)^2 = F( x^2/(1-2*x) ), where F(x) = x*M(x) and M(x) = 1 + x*M(x) + x^2*M(x)^2 is the Motzkin function (A001006).
Compare to: G(x)^2 = G( x^2/(1-2*x)^2 ), where G(x) = x*C(x)^2 and C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

Examples

			G.f.: A(x) = x + 5*x^2 + 20*x^3 + 90*x^4 + 470*x^5 + 2566*x^6 + 13885*x^7 + 74435*x^8 + 400530*x^9 + 2183930*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^5 ).
RELATED SERIES.
A(x)^2 = x^2 + 10*x^3 + 65*x^4 + 380*x^5 + 2240*x^6 + 13432*x^7 + 80330*x^8 + 474960*x^9 + 2783590*x^10 + ...
(A(x)/x)^(1/5) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 205*x^5 + 989*x^6 + 4785*x^7 + 23881*x^8 + 124245*x^9 + 673020*x^10 + ...
x/Series_Reversion( A( x^5/(1-2*x)^5 )^(1/5) ) = 1 + 2*x + x^5 - 3*x^10 + 18*x^15 - 124*x^20 + 925*x^25 - 7372*x^30 + 61466*x^35 - 528678*x^40 + 4656736*x^45 + ...
SPECIFIC VALUES.
A(t) = 3/4 at t = 0.1535987460670222421700476984635848015956844093413...
A(t) = 3/5 at t = 0.1494534252284609931621062683479802340037678508370...
A(t) = 1/2 at t = 0.1443598468225794843508026942502138500132562159005...
A(t) = 2/5 at t = 0.1362812665991487577089709044456104123756230678872...
A(t) = 1/4 at t = 0.1144692674833411472616636812900607840273720167873...
A(1/7) = 0.477612316813393143429515106540189409592882329142...
where A(1/7)^2 = A(343/3125).
A(1/8) = 0.309560069127977498956512592550239740786137843207...
where A(1/8)^2 = A(16/243).
A(1/9) = 0.234151149075763124751821214511435118422621268792...
where A(1/9)^2 = A(729/16807).
A(1/10) = 0.189302960006249918030251616127177047165765112599...
where A(1/10)^2 = A(125/4096).

Links

Crossrefs

Cf. A001006, A000108, A375453, A375454, A375456.

Programs

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

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2/(1-2*x)^5 ).
(2) A(x)^4 = A( x^4*(1-2*x)^15 / ((1-2*x)^5 - 2*x^2)^5 ).
(3) A(x^2*(1 + 2*x)^3) = A( x/(1+2*x) )^2.
The radius of convergence r satisfies r = (1 - 2*r)^5, where A(r) = 1 and r = 0.155430268881057887439965...
Showing 1-3 of 3 results.

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

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