A214372 -id:A214372 - cp's OEIS frontend

A361305 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^3.

1, 1, 5, 23, 123, 700, 4170, 25677, 162101, 1043603, 6825429, 45222437, 302892681, 2047499720, 13950769772, 95710823820, 660609751890, 4584018016679, 31960334260971, 223782306725768, 1572921720684820, 11094267854522250, 78499108540111380, 557041048588402170

Offset: 1

Paul D. Hanna, Mar 08 2023

nonn

			G.f.: A(x) = x + x^2 + 5*x^3 + 23*x^4 + 123*x^5 + 700*x^6 + 4170*x^7 + 25677*x^8 + 162101*x^9 + 1043603*x^10 + ...
such that A(x) = x + A(x)^2 * (1 + A(x))^3.
Related series.
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 56*x^5 + 317*x^6 + 1876*x^7 + 11499*x^8 + 72352*x^9 + 464585*x^10 + ...
(1 + A(x))^3 = 1 + 3*x + 6*x^2 + 22*x^3 + 105*x^4 + 555*x^5 + 3151*x^6 + 18735*x^7 + 115200*x^8 + 726530*x^9 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A361304, A361306, A214372.

{a(n)=polcoeff(serreverse(x-x^2*(1+x)^3+x*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(3*m)/m!)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*(1+x+x*O(x^n))^(3*m)/m!))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x) = Series_Reversion( x - x^2*(1+x)^3 ).
(2) A(x) = x + A(x)^2*(1 + A(x))^3.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * (1+x)^(3*n) / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(3*n) / n! ).
(5) A(x) = x + Series_Reversion( Series_Reversion( x*(1+x)^(3/2) ) - x^2 )^2.
From Vaclav Kotesovec, Mar 09 2023: (Start)
Recurrence: 283*(n-3)*(n-2)*(n-1)*n*(3869140*n^3 - 39431172*n^2 + 133221959*n - 149076999)*a(n) = 4*(n-3)*(n-2)*(n-1)*(1199433400*n^4 - 14022813420*n^3 + 59620648652*n^2 - 107988096753*n + 68872774500)*a(n-1) + 6*(n-3)*(n-2)*(3718243540*n^5 - 52766330452*n^4 + 294066223701*n^3 - 803084308634*n^2 + 1072900001465*n - 559958090580)*a(n-2) + 12*(n-3)*(2453034760*n^6 - 43397123748*n^5 + 316599139024*n^4 - 1218191215329*n^3 + 2605017314614*n^2 - 2932345787601*n + 1355713586640)*a(n-3) + 5*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(3869140*n^3 - 27823752*n^2 + 65967035*n - 51417072)*a(n-4).
a(n) ~ 1/(2 * sqrt(Pi*(1 + s)*(1 + 8*s + 10*s^2)) * n^(3/2) * r^(n - 1/2)), where r = 0.1321273811013026086255933373480102325835852282463... and s = 0.2180852364825231879900920777342190033594997222087... are real roots of the system of equations r + s^2 * (1+s)^3 = s, s * (1+s)^2 * (2+5*s) = 1. (End)
a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(3*k,n-k)/(n+k+1). - Seiichi Manyama, Aug 24 2023

A361306 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^4.

Original entry on oeis.org

1, 1, 6, 31, 186, 1191, 7972, 55164, 391322, 2830751, 20801826, 154853413, 1165316224, 8850372878, 67750780816, 522218420336, 4049564739054, 31570368061361, 247293510244174, 1945331619223591, 15361731119713506, 121729460653957980, 967664450692965300

Offset: 1

Paul D. Hanna, Mar 08 2023

nonn

			G.f.: A(x) = x + x^2 + 6*x^3 + 31*x^4 + 186*x^5 + 1191*x^6 + 7972*x^7 + 55164*x^8 + 391322*x^9 + ...
such that sqrt(A(x) - x) = A(x)*(1 + A(x))^2.
A(x)*(1 + A(x))^2 = x + 3*x^2 + 11*x^3 + 60*x^4 + 355*x^5 + 2261*x^6 + 15094*x^7 + 104208*x^8 + ...
A(x)*(1 + A(x))^2 = Series_Reversion( -x^2 + Sum_{n>=1} (-1)^(n-1) * binomial(3*n-2,n-1)*x^n/n ).

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A361304, A361305, A214372, A006013.

{a(n)=polcoeff(serreverse(x - x^2*(1+x)^4 +x*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(4*m)/m!)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*(1+x+x*O(x^n))^(4*m)/m!))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x) = Series_Reversion( x - x^2*(1+x)^4 ).
(2) A(x) = x + A(x)^2 * (1 + A(x))^4.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * (1+x)^(4*n) / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(4*n) / n! ).
(5) A(x) = x + Series_Reversion( Series_Reversion( x*(1+x)^2 ) - x^2 )^2.
(6) A(x) = x + Series_Reversion( -x^2 + Sum_{n>=1} (-1)^(n-1) * binomial(3*n-2,n-1) * x^n/n )^2.
From Vaclav Kotesovec, Mar 09 2023: (Start)
Recurrence: 3381*(n-4)*(n-3)*(n-2)*(n-1)*n*(4485934293448*n^5 - 88905588075732*n^4 + 698950092208066*n^3 - 2724285958475163*n^2 + 5263801532363671*n - 4032831805999290)*a(n) = 2*(n-4)*(n-3)*(n-2)*(n-1)*(33204885640102096*n^6 - 707886491396721408*n^5 + 6160367858867908768*n^4 - 27918165429184721124*n^3 + 69150795811214975011*n^2 - 88077097294043237943*n + 44480953779348451050)*a(n-1) + 8*(n-4)*(n-3)*(n-2)*(52772531028122272*n^7 - 1256975462235400336*n^6 + 12611049851568548176*n^5 - 69004162305753446968*n^4 + 222104765912229832762*n^3 - 419924105934755620321*n^2 + 431120275047208552290*n - 185089750933520270250)*a(n-2) + 48*(n-4)*(n-3)*(17647665510424432*n^8 - 482112074818112928*n^7 + 5693971809001104840*n^6 - 37956706633792772384*n^5 + 156126872715173363823*n^4 - 405548028261835673882*n^3 + 649232078072133939050*n^2 - 585187986606994739801*n + 227161430445970883100)*a(n-3) + 32*(n-4)*(21945190563547616*n^9 - 698268423629052336*n^8 + 9788485232517982416*n^7 - 79313303231764021176*n^6 + 409187506797434806734*n^5 - 1393249646753024170299*n^4 + 3129189249705937191544*n^3 - 4467594298222926610959*n^2 + 3676695031470911619960*n - 1327813620065788842000)*a(n-4) + 72*(2*n - 7)*(3*n - 14)*(3*n - 13)*(6*n - 31)*(6*n - 29)*(4485934293448*n^5 - 66475916608492*n^4 + 388187082839618*n^3 - 1116009867370877*n^2 + 1578887211201855*n - 878785793685000)*a(n-5).
a(n) ~ 1/(2 * (1 + s) * sqrt(Pi*(1 + 10*s + 15*s^2)) * n^(3/2) * r^(n - 1/2)), where r = 0.1176087332021218420455915375218722861407778043565... and s = 0.1894485384658193296593809633217117092941452563863... are real roots of the system of equations r + s^2 * (1+s)^4 = s, 2*s*(1+s)^3 * (1+3*s) = 1. (End)
a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(4*k,n-k)/(n+k+1). - Seiichi Manyama, Aug 24 2023

A215128 G.f.: Sum_{n>=0} d^n/dx^n (x + x^2)^(2*n) / n!.

Original entry on oeis.org

1, 2, 12, 64, 370, 2184, 13132, 79944, 491238, 3040400, 18926336, 118369368, 743199184, 4681668488, 29574616440, 187281906512, 1188494457492, 7556371963488, 48123031011036, 306929964849200, 1960230225450420, 12534313062502440, 80236414444623240

Offset: 0

Paul D. Hanna, Aug 04 2012

nonn

Compare to: Sum_{n>=0} d^n/dx^n x^(2*n)/n! = 1/sqrt(1-4*x).

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 64*x^3 + 370*x^4 + 2184*x^5 + 13132*x^6 +...
such that, by definition:
A(x) = 1 + d/dx (x+x^2)^2 + d^2/dx^2 (x+x^2)^4/2! + d^3/dx^3 (x+x^2)^6/3! + d^4/dx^4 (x+x^2)^8/4! + d^5/dx^5 (x+x^2)^10/5! +...

Cf. A214372, A215125, A215129, A038112.

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

a(n) = (n+1)*A214372(n+1), where G(x) = x + (G(x) + G(x)^2)^2 is the g.f. of A214372.

A229042 Series reversion of (sqrt(1+4*x) - 1)/2 - x^2.

Original entry on oeis.org

1, 2, 6, 25, 114, 560, 2880, 15321, 83600, 465322, 2631668, 15079922, 87362184, 510837760, 3010987912, 17870854206, 106713713826, 640659094566, 3864643224900, 23412690485800, 142386465217920, 868967571732540, 5320093500153120, 32666064906596550, 201109232686971492

Offset: 1

Paul D. Hanna, Oct 24 2013

nonn

			G.f.: A(x) = x + 2*x^2 + 6*x^3 + 25*x^4 + 114*x^5 + 560*x^6 + 2880*x^7 +...
where the series reversion of g.f. A(x) begins:
(sqrt(1+4*x) - 1)/2 - x^2 = x - 2*x^2 + 2*x^3 - 5*x^4 + 14*x^5 - 42*x^6 + 132*x^7 - 429*x^8 +...+ (-1)^(n-1)*A000108(n-1)*x^n +...
The square of the g.f. equals the series:
A(x)^2 = x^2*(1+x)^2 + d/dx x^4*(1+x)^4/2! + d^2/dx^2 x^6*(1+x)^6/3! + d^3/dx^3 x^8*(1+x)^8/4! + d^4/dx^4 x^10*(1+x)^10/5! +...
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 + 9993*x^8 +...
Related Expansions:
G.f. A(x) = G(x) + G(x)^2 = sqrt(G(x) - x) where G(x) is the g.f. of A214372:
G(x) = x + x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 + 9993*x^8 +...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A214372, A229043.

Rest[CoefficientList[InverseSeries[Series[(Sqrt[1+4*x]-1)/2-x^2,{x,0,20}],x],x]] (* Vaclav Kotesovec, Jan 22 2014 *)

a(n):=binomial(2*(n-1),(n-1))/n-sum(k*(-1)^(k-n)*binomial(n+k-1,n-1)*((sum((binomial(k-1,l)*binomial(2*(n+l-k),n-k-1))/(n+l-k),l,0,k-1))),k,1,n-1)/n; /* Vladimir Kruchinin, Feb 08 2015 */

{a(n)=polcoeff( serreverse( (sqrt(1+4*x +x*O(x^n)) - 1)/2 - x^2 ), n)}
for(n=1, 30, print1(a(n), ", "))

/* G.f. A(x) = sqrt(G(x) - x) where G(x) = x + G(x)^2*(1 + G(x))^2 */
{a(n)=local(G=serreverse(x-x^2*(1+x)^2+x^2*O(x^n)));polcoeff(sqrt(G-x),n)}
for(n=1, 30, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ n-th derivative
{a(n)=local(A2=x); A2=sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); polcoeff(sqrt(A2), n)}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x)^2 = Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*(1+x)^(2*n)/n!.
(2) A(x)^2 = A(x)*C(-A(x)) - x, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(3) A(x) = G(x)*(1 + G(x)) = sqrt(G(x) - x) where G(x) is the g.f. of A214372.
a(n) ~ r^(1/2-n) / (2*sqrt(Pi) * sqrt(1+8*s^3) * n^(3/2)), where s = ((3392 - 384*sqrt(78))^(1/3) + 4*(53 + 6*sqrt(78))^(1/3) - 4)/48 = 0.328649053... is the root of the equation 4*s^2*(1+4*s) = 1 and r = -1/2 + 1/(4*s) - s^2 = 0.152679653319... - Vaclav Kotesovec, Jan 22 2014
a(n) = binomial(2*(n-1),(n-1))/n-sum(k=1..n-1, k*(-1)^(k-n)*binomial(n+k-1,n-1)*((sum(l=0..k-1, (binomial(k-1,l)*binomial(2*(n+l-k),n-k-1))/(n+l-k)))))/n. - Vladimir Kruchinin, Feb 08 2015
D-finite with recurrence 3503*n*(n-1)*(n-2)*a(n) -4*(n-1)*(n-2)*(3377*n-3903)*a(n-1) -20*(n-2)*(2668*n^2-11692*n+12387)*a(n-2) +8*(-6616*n^3+57120*n^2-162464*n+152595)*a(n-3) -480*(4*n-15)*(4*n-13)*(2*n-9)*a(n-4)=0. - R. J. Mathar, Mar 24 2023

A361304 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2n) (1 + x)^(4*n) / n!.

Original entry on oeis.org

1, 2, 18, 124, 930, 7146, 55804, 441312, 3521898, 28307510, 228820086, 1858240956, 15149110912, 123905220292, 1016261712240, 8355494725376, 68842600563918, 568266625104498, 4698576694639306, 38906632384471820, 322596353513983626, 2678048134387075560

Offset: 0

Paul D. Hanna, Mar 08 2023

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 124*x^3 + 930*x^4 + 7146*x^5 + 55804*x^6 + 441312*x^7 + 3521898*x^8 + 28307510*x^9 + ...

Cf. A361306, A214372.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); A = sum(m=0, n, Dx(m, x^(2*m)*(1+x +O(x^(n+1)))^(4*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); A = deriv( serreverse(x - x^2*(1+x +O(x^(n+3)))^4 )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*(1 + x)^4).
(3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(4*n) / n! ) is the g.f. of A361306.
(4) a(n) = (n+1) * A361306(n+1) for n >= 0.

A367282 G.f. satisfies A(x) = 1 + xA(x)^2 (1 + x*A(x)^2)^2.

Original entry on oeis.org

1, 1, 4, 18, 94, 527, 3108, 18993, 119214, 763997, 4978304, 32883853, 219690066, 1481858835, 10078051830, 69030877581, 475795428158, 3297527987794, 22965847261928, 160649189379029, 1128201207643744, 7951399289858530, 56222323349767666

Offset: 0

Seiichi Manyama, Nov 12 2023

nonn

Cf. A002293, A073155, A214372, A367283.
Cf. A000108, A367258.
Cf. A367237.

a(n, s=2, t=2, u=2) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).

A365155 G.f. satisfies A(x) = ( 1 + xA(x)^2(1 + x*A(x))^2 )^2.

Original entry on oeis.org

1, 2, 13, 98, 838, 7690, 74047, 738028, 7549658, 78811732, 836219773, 8991739874, 97769604542, 1073156173442, 11875174074608, 132333387616600, 1483789788291516, 16727705523572128, 189496296040063170, 2155984626357225948, 24625450759174328948

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A214372, A365156.

a(n, s=2, t=2) = sum(k=0, n, binomial(t*(n+k+1), k)*binomial(s*k, n-k)/(n+k+1));

If g.f. satisfies A(x) = ( 1 + x*A(x)^2*(1 + x*A(x))^s )^t, then a(n) = Sum_{k=0..n} binomial(t*(n+k+1),k) * binomial(s*k,n-k)/(n+k+1).

A365156 G.f. satisfies A(x) = ( 1 + xA(x)^2(1 + x*A(x))^2 )^3.

Original entry on oeis.org

1, 3, 27, 295, 3648, 48513, 677450, 9797031, 145458252, 2204380144, 33960095667, 530268482913, 8373331428836, 133484219528982, 2145376940485452, 34725549386905863, 565567039020594492, 9261756210015412356, 152410211630410153468

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A214372, A365155.

a(n, s=2, t=3) = sum(k=0, n, binomial(t*(n+k+1), k)*binomial(s*k, n-k)/(n+k+1));

If g.f. satisfies A(x) = ( 1 + x*A(x)^2*(1 + x*A(x))^s )^t, then a(n) = Sum_{k=0..n} binomial(t*(n+k+1),k) * binomial(s*k,n-k)/(n+k+1).

A367259 G.f. satisfies A(x) = 1 + xA(x)^3 (1 + x*A(x))^2.

Original entry on oeis.org

1, 1, 5, 27, 169, 1138, 8061, 59188, 446455, 3438863, 26935372, 213883631, 1717852129, 13931065117, 113913095218, 938154381748, 7774936633411, 64791892224825, 542598513709481, 4564001359135661, 38541714429405304, 326640923339410701

Offset: 0

Seiichi Manyama, Nov 11 2023

nonn

Cf. A007477, A161634, A214372.

Cf. A367232.

A367259 := proc(n)
    add(binomial(3*k+(n-k)+1,k) * binomial(2*k,n-k) / (3*k+(n-k)+1),k=0..n) ;
end proc:
seq(A367259(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

a(n, s=2, t=3, u=1) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).

D-finite with recurrence 8*n*(26345215448853860010445423574*n -20961990363613887876514780359) *(2*n+1)*(2*n-1) *(n+1)*a(n) +8*n*(2*n-1)* (52690430897707720020890847148*n^3 -8177454564962587489822763646077*n^2 +19278991143331529980160099092658*n-11257584930903257675329694643457) *a(n-1) +2*(-268110383402413819740981254825038*n^5 +2815977437639263120434136294300085*n^4 -10349136726006717489413692948200650*n^3 +17659039091779726381787370980047525*n^2 -14385155927699861644653059971375422*n +4528097093401255127907905744957880) *a(n-2) +2*(2433809541139490470204589489378644*n^5 -29695021140710269817720089645612595*n^4 +147295722233051282998786410783007430*n^3 -369735985683645289967183608338045205*n^2 +467786753736867474630837654962591406*n -237792800129696483300250545739991320) *a(n-3) +2*(-2771166843885660051994398777044296*n^5 +70669385063622159693270493531099173*n^4 -615983650096141972053534317661369592*n^3 +2483715780351994976504831765723882733*n^2 -4785455586973998561063713309602358866*n +3584098048545781487176463022333484200) *a(n-4) +(-9719685405660460345432742140418255*n^5 +101488259196839193588678566387929584*n^4 +90729575616085725241944658061815579*n^3 -4613811089886954307541928620224211376*n^2 +19004407111946953332012410754931442164*n -24034005967022354223275806054437127680) *a(n-5) +5*(4577999937824616490204866357596875*n^5 -139866352876176382080407833814299250*n^4 +1615537655758910403049946493111918725*n^3 -8980705919938192198141139371077714070*n^2 +24274376174445463863335689528941329496*n -25684691566228873512769902557616240960) *a(n-6) +30*(1336493817891495200869338759185*n -5552932550849126027962496889033) *(5*n-32)*(5*n-26)*(5*n-29)*(5*n-28)*a(n-7)=0. - R. J. Mathar, Dec 04 2023

A367283 G.f. satisfies A(x) = 1 + xA(x)^2 (1 + x*A(x)^3)^2.

Original entry on oeis.org

1, 1, 4, 20, 116, 728, 4818, 33100, 233824, 1687764, 12393520, 92291681, 695325926, 5290359124, 40591599128, 313725215636, 2440203573816, 19087022233906, 150042056387660, 1184734863936672, 9392213303130904, 74728563957003952, 596531545003840160

Offset: 0

Seiichi Manyama, Nov 12 2023

nonn

Cf. A002293, A073155, A214372, A367282.

Cf. A001764, A137952.

a(n, s=2, t=2, u=3) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).

cp's OEIS Frontend

A361305 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^3.

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A361306 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^4.

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A215128 G.f.: Sum_{n>=0} d^n/dx^n (x + x^2)^(2*n) / n!.

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A229042 Series reversion of (sqrt(1+4*x) - 1)/2 - x^2.

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A361304 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!.

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A367282 G.f. satisfies A(x) = 1 + x*A(x)^2 * (1 + x*A(x)^2)^2.

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A365155 G.f. satisfies A(x) = ( 1 + x*A(x)^2*(1 + x*A(x))^2 )^2.

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A365156 G.f. satisfies A(x) = ( 1 + x*A(x)^2*(1 + x*A(x))^2 )^3.

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A361304 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2n) (1 + x)^(4*n) / n!.

A367282 G.f. satisfies A(x) = 1 + xA(x)^2 (1 + x*A(x)^2)^2.

A365155 G.f. satisfies A(x) = ( 1 + xA(x)^2(1 + x*A(x))^2 )^2.

A365156 G.f. satisfies A(x) = ( 1 + xA(x)^2(1 + x*A(x))^2 )^3.

A367259 G.f. satisfies A(x) = 1 + xA(x)^3 (1 + x*A(x))^2.

A367283 G.f. satisfies A(x) = 1 + xA(x)^2 (1 + x*A(x)^3)^2.