A087949 -id:A087949 - cp's OEIS frontend

A143500 G.f. A(x) satisfies A(x) = 1 + xA(xA(x)^2).

1, 1, 1, 3, 10, 46, 244, 1481, 10020, 74400, 599573, 5200284, 48223360, 475557054, 4965035754, 54672110310, 632853655686, 7678552433184, 97404631390960, 1288861146261679, 17752479062092470, 254051633672160696, 3770953046565933003, 57964955567444706668

Offset: 0

Paul D. Hanna, Aug 20 2008

nonn

			G.f. A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 46*x^5 + 244*x^6 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 27*x^4 + 118*x^5 + 609*x^6 +...
A(x*A(x)^2) = 1 + x + 3*x^2 + 10*x^3 + 46*x^4 + 244*x^5 +...
If G(x*A(x)^2) = x then
G(x) = x - 2*x^2 + 5*x^3 - 18*x^4 + 68*x^5 - 300*x^6 + 1283*x^7 -+...
A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/2) where
A(x)*G(x) = x - x^2 + 4*x^3 - 12*x^4 + 59*x^5 - 209*x^6 + 1199*x^7 -...

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Cf. A087949, A143501.

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^2));polcoeff(A,n)}

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, j))); \\ Seiichi Manyama, Jun 04 2025

G.f. satisfies: G(x) = x/[1 + A(x)*G(x)]^2 = x/A(G(x))^2 where G(x*A(x)^2) = x.
From Seiichi Manyama, Jun 04 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+k,j)/(2*n-2*j+k) * a(n-j,j). (End)

A182969 G.f. satisfies: A(x) = 1 + xA(x)^3A(x*A(x)).

Original entry on oeis.org

1, 1, 4, 23, 159, 1236, 10454, 94401, 899286, 8964253, 92961432, 998600238, 11075132605, 126489183013, 1484601117235, 17876874457054, 220546820252773, 2784446513061287, 35940592329823310, 473893641259375150

Offset: 0

Paul D. Hanna, Dec 18 2010

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 159*x^4 + 1236*x^5 +...
Related expansions:
A(x*A(x)) = 1 + x + 5*x^2 + 35*x^3 + 287*x^4 + 2592*x^5 + 25050*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 94*x^3 + 675*x^4 + 5331*x^5 + 45274*x^6 +...
Logarithmic series:
log(A(x)) = x*A(x)^2 + [d/dx x^3*A(x)^2]*A(x)^2/2! + [d^2/dx^2 x^5*A(x)^3]*A(x)^3/3! + [d^3/dx^3 x^7*A(x)^4]*A(x)^4/4! +...

Cf. A139702, A143426, A087949, A143435.

T(n,m):=if n=m then 1 else m/n*sum(sum(T(n-m,i)*k/i*binomial(2*i-k-1,i-1),i,k,n-m)*binomial(n+k-1,n-1),k,1,n-m); makelist(T(n,1),n,1,10); /* Vladimir Kruchinin, May 07 2012 */

/* n-th Derivative: */
{Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D}
/* G.f.: */
{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,
A=exp(sum(m=0,n,Dx(m,x^(2*m+1)*A^(m+1))*A^(m+1)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}

G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(n+1)]*A(x)^(n+1)/(n+1)! ).
a(n) = T(n-1,1), where T(n,m) = (m/n)*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i)*k/i*binomial(2*i-k-1,i-1))*binomial(n+k-1,n-1)), n>m, T(n,n)=1. [Vladimir Kruchinin, May 07 2012]
T(n,m) = m * sum(k=1..n-m, (T(n-m,k)*binomial(n+2*k-1,n+k-1))/(n+k)) for n>m, and T(n,n) = 1. [Vladimir Kruchinin, Aug 08 2012]

A381570 G.f. A(x) satisfies A(x) = (1 + xA(xA(x)))^3.

Original entry on oeis.org

1, 3, 12, 82, 732, 7944, 99156, 1381464, 21065853, 346932822, 6112226961, 114383442888, 2261347164766, 47025363829497, 1025005545866361, 23349137897005296, 554467427766694440, 13696046757037152183, 351231525904387758222, 9335221780768641038952

Offset: 0

Seiichi Manyama, Feb 28 2025

nonn

Column k=1 of A381569.

Cf. A087949, A120972, A212029, A381568, A381574.

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+3*k, j)/(n-j+k)*a(n-j, j)));

See A381569.

G.f.: B(x)^3, where B(x) is the g.f. of A212029.

A381568 G.f. A(x) satisfies A(x) = (1 + xA(xA(x)))^2.

Original entry on oeis.org

1, 2, 5, 22, 126, 884, 7149, 64688, 641836, 6888740, 79203860, 968503090, 12525131474, 170555767116, 2436592516874, 36409825487380, 567612675812796, 9211031425896752, 155283809480528788, 2714788300934206360, 49140787009610861896, 919625415852055598804, 17768937720619971300781

Offset: 0

Seiichi Manyama, Feb 28 2025

nonn

Column k=1 of A381567.

Cf. A087949, A120970, A143508, A381570.

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+2*k, j)/(n-j+k)*a(n-j, j)));

See A381567.

G.f.: B(x)^2, where B(x) is the g.f. of A143508.

A384574 G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^4).

Original entry on oeis.org

1, 1, 1, 5, 23, 155, 1236, 11286, 116333, 1329433, 16630343, 225606826, 3294976854, 51496560764, 856858516809, 15112857079891, 281479726839851, 5517842789917283, 113510479973132860, 2444032094604379100, 54948814775692303024, 1287258966133883349701

Offset: 0

Seiichi Manyama, Jun 04 2025

nonn

Column k=1 of A384582.
Cf. A087949, A143500, A143501, A384575.
Cf. A384578.

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-4*j+k, j)/(4*n-4*j+k)*a(n-j, j)));

See A384582.

A384575 G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^5).

Original entry on oeis.org

1, 1, 1, 6, 31, 236, 2166, 22722, 269889, 3567412, 51765431, 816476196, 13892821878, 253442895075, 4930644856063, 101830536332051, 2223767436058566, 51172807259226084, 1237092039069090235, 31332521053777095784, 829389782837272248191, 22894754438382163120136

Offset: 0

Seiichi Manyama, Jun 04 2025

nonn

Column k=1 of A384583.
Cf. A087949, A143500, A143501, A384574.
Cf. A384579.

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n-5*j+k, j)/(5*n-5*j+k)*a(n-j, j)));

See A384583.

A087959 G.f. satisfies A(x) = 1 + xA(xA(x*A(x))).

Original entry on oeis.org

1, 1, 1, 2, 6, 22, 95, 463, 2483, 14426, 89799, 594038, 4150514, 30482791, 234428282, 1881944298, 15729230166, 136566166388, 1229346044429, 11454129882773, 110293243452027, 1096078120808889, 11227947692444477, 118421577443291274, 1284625091089249880

Offset: 0

Paul D. Hanna, Sep 16 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A087949.

A:= proc(n) option remember; local T; if n=0 then 1 else T:= A(n-1); unapply(convert(series(1+ x*T(x*T(x*T(x))), x, n+1), polynom), x) fi end: a:= n-> coeff(A(n)(x), x, n): seq(a(n), n=0..21);  # Alois P. Heinz, Aug 23 2008

A[n_] := A[n] = Module[{T}, If[n == 0, 1, T[y_] = A[n-1][y]; Normal[Series[1+x*T[x*T[x*T[x]]], {x, 0, n+1}]] /. x -> #]&]; a[n_] := Coefficient[A[n][x], x, n]; Table [a[n], {n, 0, 24}] (* Jean-François Alcover, Feb 14 2014, after Alois P. Heinz *)

A140092 G.f. satisfies: A(x) = Series_Reversion[ x/sqrt(1 + 4*A(x)) ] with A(0)=0.

Original entry on oeis.org

1, 2, 6, 28, 174, 1308, 11300, 108808, 1145078, 12996332, 157580252, 2026874424, 27507762028, 392226116696, 5855551243464, 91263899531280, 1481385005886374, 24989341719984972, 437270678940944556, 7923785627972483672

Offset: 1

Paul D. Hanna, May 15 2008

nonn

			G.f.: A(x) = x +2*x^2 +6*x^3 + 28*x^4 + 174*x^5 +1308*x^6 +11300*x^7 +...
A(A(x)) = x + 4*x^2 + 20*x^3 +124*x^4 + 912*x^5 +7676*x^6 +72064*x^7 +...
A(x)^2 = x^2 +4*x^3 + 16*x^4 + 80*x^5 + 496*x^6 +3648*x^7 +30704*x^8 +...

Cf. A087949.

array(TL, fixnum, 30, 30); T(n, m):=if n=m then 1 else if TL[n, m]=0 then TL[n, m]:m/n*sum(sum(T(n-m, i)*sum(binomial(-k-j+2*i-1,i-1)*(k+j)*2^(k+j)*binomial(k+j-1,k-1),j,0,i-k)/i*(-1)^(i+k), i, k, n-m)*binomial(n+k-1, n-1), k, 1, n-m) else TL[n, m]; makelist(T(n,1),n,1,15); /* Vladimir Kruchinin, May 08 2012 */

{a(n)=local(A=x); if(n<1, 0, for(i=1,n,A=serreverse(x/sqrt(1+4*A +x*O(x^n)))); polcoeff(A, n))}

G.f. satisfies: A(x) = x*sqrt(1 + 4*A(A(x))).
G.f. satisfies: A(A(x)) = (A(x)^2 - x^2)/(2*x)^2.
a(n)=T(n,1), T(n, m)=m/n*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i)*sum(j=0..i-k, binomial(-k-j+2*i-1,i-1)*(k+j)*2^(k+j)*binomial(k+j-1,k-1))/i*(-1)^(i+k))*binomial(n+k-1, n-1)), n>m, T(n,n)=1. - Vladimir Kruchinin, May 08 2012

A381649 G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 * A(x*A(x)^3)^3.

Original entry on oeis.org

1, 1, 5, 44, 510, 7024, 109362, 1871530, 34590180, 682396379, 14251399805, 313170119013, 7207845252630, 173129413258492, 4327373963163746, 112289379643018983, 3018922654575996866, 83951253980821314446, 2411137697712963195801, 71427857356498491780290

Offset: 0

Seiichi Manyama, Mar 03 2025

nonn

Column k=1 of A381648.
Cf. A120973, A212029, A381601, A381615.
Cf. A087949, A381029.

a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-j+k, j)/(3*n-j+k)*a(n-j, 3*j)));

See A381648.

cp's OEIS Frontend

A143500 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2).

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A182969 G.f. satisfies: A(x) = 1 + x*A(x)^3*A(x*A(x)).

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A381570 G.f. A(x) satisfies A(x) = (1 + x*A(x*A(x)))^3.

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A381568 G.f. A(x) satisfies A(x) = (1 + x*A(x*A(x)))^2.

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A384574 G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^4).

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A384575 G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^5).

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PARI

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A087959 G.f. satisfies A(x) = 1 + x*A(x*A(x*A(x))).

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Maple

Mathematica

A140092 G.f. satisfies: A(x) = Series_Reversion[ x/sqrt(1 + 4*A(x)) ] with A(0)=0.

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Maxima

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A381649 G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 * A(x*A(x)^3)^3.

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A143500 G.f. A(x) satisfies A(x) = 1 + xA(xA(x)^2).

A182969 G.f. satisfies: A(x) = 1 + xA(x)^3A(x*A(x)).

A381570 G.f. A(x) satisfies A(x) = (1 + xA(xA(x)))^3.

A381568 G.f. A(x) satisfies A(x) = (1 + xA(xA(x)))^2.

A087959 G.f. satisfies A(x) = 1 + xA(xA(x*A(x))).