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.

A375453 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^3 ).

Original entry on oeis.org

1, 3, 9, 31, 117, 459, 1835, 7449, 30711, 128601, 546537, 2354139, 10260492, 45173868, 200578692, 896865572, 4033380894, 18224524458, 82664886074, 376161628302, 1716301466139, 7848924260901, 35966629306221, 165109474283847, 759210907786198, 3496438156668822, 16126158739138860
Offset: 1

Views

Author

Paul D. Hanna, Aug 16 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 + 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 ).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 27*x^4 + 116*x^5 + 501*x^6 + 2178*x^7 + 9491*x^8 + 41424*x^9 + 181293*x^10 + ...
(A(x)/x)^(1/3) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 77*x^5 + 290*x^6 + 1122*x^7 + 4462*x^8 + 18210*x^9 + ... + A375443(n)*x^n + ...
x/Series_Reversion( A( x^2/(1-2*x) )^(1/2) ) = 1 + x + 2*x^2 - 2*x^4 + 6*x^6 - 20*x^8 + 70*x^10 - 263*x^12 + 1044*x^14 - 4263*x^16 + 17762*x^18 + ...
x/Series_Reversion( A( x^3/(1-2*x)^3 )^(1/3) ) = 1 + 2*x + x^3 - x^6 + 3*x^9 - 10*x^12 + 34*x^15 - 124*x^18 + 482*x^21 - 1931*x^24 + 7893*x^27 + ...
SPECIFIC VALUES.
A(t) = 3/4 at t = 0.201772636312778304679687617697508690090653188...
A(t) = 3/5 at t = 0.194614960496736155296642077884228463225576089...
A(t) = 1/2 at t = 0.186135869221980538627401571340819246192140850...
A(t) = 2/5 at t = 0.173143830263370608074654087902797631449309857...
A(t) = 1/4 at t = 0.140069990039210460387276300843591158073987855...
A(1/5) = 0.700768312277362449514797370811301885385349818...
where A(1/5)^2 = A(5/27).
A(1/6) = 0.362320684925221039201199651574198595785551012...
where A(1/6)^2 = A(6/64).
A(1/7) = 0.259569089568076471080673806323871020166140312...
where A(1/7)^2 = A(7/125).
A(1/10) = 0.14404022241542053703979110789205898915122135...
where A(1/10)^2 = A(10/512).

Links

Crossrefs

Cf. A001006, A000108, A375454, A375455, A375456.
Cf. A375443.

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)^3 ) - 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)^3 ).
(2) A(x)^4 = A( x^4*(1-2*x)^3 / ((1-4*x)^3*(1 - 2*x + 2*x^2)^3) ).
(3) A(x^2 + 2*x^3) = A( x/(1+2*x) )^2.
The radius of convergence r satisfies r = (1 - 2*r)^3, where A(r) = 1 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...

A375445 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).

Original entry on oeis.org

1, 1, 2, 8, 41, 205, 989, 4785, 23881, 124245, 673020, 3771678, 21702164, 127311556, 756930002, 4539680854, 27367146987, 165407567379, 1000581963363, 6051411131431, 36569087782730, 220760294880122, 1331294835476618, 8021165000866546, 48296514171243436, 290695754850732916
Offset: 0

Views

Author

Paul D. Hanna, Aug 19 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) = 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 + ...
where A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 102*x^4 + 524*x^5 + 2616*x^6 + 13024*x^7 + 66249*x^8 + 348026*x^9 + 1889737*x^10 + ...
A(x)^5 = 1 + 5*x + 20*x^2 + 90*x^3 + 470*x^4 + 2566*x^5 + 13885*x^6 + 74435*x^7 + 400530*x^8 + ... + A375455(n+1)*x^n + ...
SPECIFIC VALUES.
Given the radius of convergence r = 0.15543026888105788743996...,
A(r) = 1.4510850920547193207944317544312912656627353873916...
  where r = (1-2*r)^5 and A(r) = 1/(1-2*r).
A(1/7) = 1.273018489928554436323320513425747043274176403249...
  where A(1/7)^2 = (7/5)*A(343/3125).
A(1/8) = 1.198855898496093050319216983995020709132914678012...
  where A(1/8)^2 = (4/3)*A(16/243).
A(1/9) = 1.160774237134743051625929742274648689798420066384...
  where A(1/9)^2 = (9/7)*A(729/16807).
A(1/10) = 1.136139033822992899751347322772302396437733019439...
  where A(1/10)^2 = (5/4)*A(125/4096).

Links

Crossrefs

Cf. A001006, A000108, A375455, A375443, A375444.

Programs

  • Mathematica
    terms = 26; A[] = 1; Do[A[x] = Sqrt[A[x^2 /(1 - 2x)^5]/(1 - 2x)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Aug 11 2025 *)
  • 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)^5 )/(1-2*x) - Ax^2, #A-1) ); 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)^5 )/(1-2*x).
(2) A(x)^4 = A( x^4*y^5 )*y where y = (1-2*x)^3/((1-2*x)^5 - 2*x^2).
(3) A( x^2*(1 + 2*x)^3 ) = A( x/(1+2*x) )^2 / (1+2*x).
The radius of convergence r satisfies r = (1 - 2*r)^5, where A(r) = 1/(1-2*r) and r = 0.1554302688810578874399658483538386517334...
Showing 1-3 of 3 results.

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