A369215 -id:A369215 - cp's OEIS frontend

A379172 G.f. A(x) satisfies A(x) = (1 + xA(x)^3)/(1 - xA(x))^3.

1, 4, 33, 358, 4445, 59745, 846023, 12430941, 187753479, 2896929975, 45465112431, 723520554096, 11647721390271, 189352106241567, 3104046096391902, 51254005259550753, 851674902290491936, 14231191062537888864, 238978853442142491358, 4030889937027642017872

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A365764, A369215, A379171.

a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(4*n+5*k+2, n-k)/(n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(4*n+5*k+2,n-k)/(n+2*k+1).

A379171 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^3.

Original entry on oeis.org

1, 4, 21, 139, 1021, 8010, 65708, 556751, 4834686, 42800265, 384832083, 3504693519, 32261240127, 299685628629, 2805773759322, 26448278629697, 250806022116194, 2390973659474304, 22901157688878983, 220279614235505630, 2126890041331033797, 20606993367985131716

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A365764, A369215, A379172.

Cf. A025227, A379173.

a(n) = sum(k=0, n, binomial(n-k+1, k)*binomial(4*n-4*k+2, n-k)/(n-k+1));

a(n) = Sum_{k=0..n} binomial(n-k+1,k) * binomial(4*n-4*k+2,n-k)/(n-k+1).

A369216 Expansion of (1/x) * Series_Reversion( x * ((1-x)^4-x) ).

Original entry on oeis.org

1, 5, 44, 479, 5827, 75887, 1034980, 14593794, 211031650, 3112385177, 46636714566, 707983562624, 10865572966703, 168306274609798, 2627854427929448, 41314461126179272, 653481096161664690, 10391753978329136808, 166040704868503173384

Offset: 0

Seiichi Manyama, Jan 16 2024

nonn

Index entries for reversions of series

Cf. A151374, A249924, A369215.

Cf. A369102.

my(N=20, x='x+O('x^N)); Vec(serreverse(x*((1-x)^4-x))/x)

a(n) = sum(k=0, n, binomial(n+k, k)*binomial(5*n+3*k+3, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(5*n+3*k+3,n-k).

A369694 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^2) ).

Original entry on oeis.org

1, 3, 16, 106, 786, 6244, 51964, 447201, 3947306, 35538668, 325098696, 3013060258, 28232408848, 267003169668, 2545341982728, 24433290332007, 235967943943224, 2291147902820524, 22352525061549604, 219006814853751540, 2154083325737401740

Offset: 0

Seiichi Manyama, Jan 29 2024

nonn

Index entries for reversions of series

Cf. A001002, A151374.

Cf. A369114, A369161, A369215.

CoefficientList[InverseSeries[Series[x*((1-x)^3 - x^2), {x, 0, 30}], x]/x, x](* Vaclav Kotesovec, Jan 29 2024 *)

my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^2))/x)

a(n) = sum(k=0, n\2, binomial(n+k, k)*binomial(4*n+k+2, n-2*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(4*n+k+2,n-2*k).

a(n) ~ sqrt((60 + (220324 - 42734*sqrt(2))^(1/3) + (220324 + 42734*sqrt(2))^(1/3)) / (138*Pi)) * (((4/23)*(22 + 3*(293 - 92*sqrt(2))^(1/3) + 3*(293 + 92*sqrt(2))^(1/3)))^n / n^(3/2)). - Vaclav Kotesovec, Jan 29 2024

A370282 Coefficient of x^n in the expansion of 1/( (1-x)^3 - x )^n.

Original entry on oeis.org

1, 4, 42, 499, 6250, 80634, 1060269, 14127852, 190102482, 2577310285, 35150819132, 481734467955, 6628611532621, 91517611501008, 1267182734325900, 17589579427715124, 244689432718144770, 3410399867585709501, 47613678409439712861, 665756829352248572725

Offset: 0

Seiichi Manyama, Feb 13 2024

nonn

Cf. A069723, A370280.

Cf. A369215.

a(n) = sum(k=0, n, binomial(n+k-1, k)*binomial(4*n+2*k-1, n-k));

a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(4*n+2*k-1,n-k).

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * ((1-x)^3 - x) ). See A369215.

A369512 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x)^2 ).

Original entry on oeis.org

1, 8, 106, 1706, 30459, 580138, 11548831, 237408978, 5001034821, 107387829120, 2341915361920, 51727723741200, 1154821390130868, 26016595619565008, 590718564607726952, 13504019611821648448, 310553715057038411358, 7179645587769992602252

Offset: 0

Seiichi Manyama, Jan 25 2024

nonn

Index entries for reversions of series

Cf. A369215.

CoefficientList[InverseSeries[Series[x((1-x)^3-x)^2,{x,0,18}],x]/x,x] (* Stefano Spezia, Mar 31 2025 *)

my(N=20, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x)^2)/x)

a(n) = sum(k=0, n, binomial(2*n+k+1, k)*binomial(7*n+2*k+5, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(7*n+2*k+5,n-k).

A371433 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3 + x) ).

Original entry on oeis.org

1, 2, 5, 11, 11, -77, -704, -3795, -15686, -48598, -74009, 376623, 4438840, 27458060, 126898948, 440550682, 849522927, -2621906045, -39993434701, -270428078305, -1339005344985, -5014789377825, -11407684195950, 18849058485855, 417417757017612, 3058172078113944

Offset: 0

Seiichi Manyama, Mar 23 2024

sign

Index entries for reversions of series

Cf. A371434, A371435.

Cf. A369215.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3+x))/x)

a(n) = sum(k=0, n, (-1)^k*binomial(n+k, k)*binomial(4*n+2*k+2, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(n+k,k) * binomial(4*n+2*k+2,n-k).

A379187 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x))^3).

Original entry on oeis.org

1, 4, 30, 286, 3091, 36063, 442898, 5642628, 73893561, 988585443, 13453580815, 185661101085, 2592069904059, 36545520229810, 519601325300487, 7441580996167052, 107255985242888943, 1554576968046707916, 22644622298400113411, 331322620547205661043

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A118971, A369617, A379188.

Cf. A369215.

a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(4*n+2*k+2, n-k)/(n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(4*n+2*k+2,n-k)/(n+2*k+1).

A379245 G.f. A(x) satisfies A(x) = ( (1 + xA(x)^2)/(1 - xA(x)) )^3.

Original entry on oeis.org

1, 6, 72, 1100, 18984, 352608, 6879152, 139012368, 2884353888, 61091682368, 1315450042368, 28709737064064, 633684940733696, 14120739728984832, 317243001537462528, 7178031348934793472, 163423203504309020160, 3741114809852278047744

Offset: 0

Seiichi Manyama, Dec 18 2024

nonn

Cf. A365843, A379246.

Cf. A363380, A369215.

a(n) = sum(k=0, n, binomial(3*n+3*k+3, k)*binomial(4*n+2*k+2, n-k)/(n+k+1));

G.f.: B(x)^3 where B(x) is the g.f. of A363380.

a(n) = Sum_{k=0..n} binomial(3*n+3*k+3,k) * binomial(4*n+2*k+2,n-k)/(n+k+1).

cp's OEIS Frontend

A379172 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3)/(1 - x*A(x))^3.

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A379171 G.f. A(x) satisfies A(x) = (1 + x)/(1 - x*A(x))^3.

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A369216 Expansion of (1/x) * Series_Reversion( x * ((1-x)^4-x) ).

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A369694 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^2) ).

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A370282 Coefficient of x^n in the expansion of 1/( (1-x)^3 - x )^n.

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A369512 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x)^2 ).

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Mathematica

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A371433 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3 + x) ).

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A379187 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^2) * (1 - x*A(x))^3).

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A379245 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^2)/(1 - x*A(x)) )^3.

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A379172 G.f. A(x) satisfies A(x) = (1 + xA(x)^3)/(1 - xA(x))^3.

A379187 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x))^3).

A379245 G.f. A(x) satisfies A(x) = ( (1 + xA(x)^2)/(1 - xA(x)) )^3.