A049140 -id:A049140 - cp's OEIS frontend

A052708 A simple context-free grammar: convolution square of A049140.

0, 0, 1, 2, 5, 16, 56, 204, 768, 2970, 11726, 47060, 191412, 787304, 3269100, 13684864, 57691353, 244713654, 1043684478, 4472828400, 19252045120, 83188965420, 360734837280, 1569296837160, 6846931211250, 29954007587556, 131367797081352, 577451514567536

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 663

Cf. A049140.

spec := [S,{C=Prod(S,S),S=Prod(B,B),B=Union(S,C,Z)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

G.f.: RootOf(-_Z+_Z^4+_Z^2+x)^2.

Recurrence: {a(1)=0, a(2)=1, a(3)=2, a(4)=5, (-576-1920*n+3072*n^2+6144*n^3)*a(n)+(-9096-30320*n-28032*n^2-7936*n^3)*a(n+1)+(41380*n+20808+28032*n^2+6272*n^3)*a(n+2)+(-26520*n^2-60704*n-45600-3784*n^3)*a(n+3)+(589*n^3+5301*n^2+15314*n+14136)*a(n+4)}.

A052710 A simple context-free grammar: difference between A049140 and its convolution square.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 14, 52, 201, 792, 3168, 12844, 52676, 218148, 910996, 3832072, 16222352, 69061200, 295477550, 1269863304, 5479456290, 23730089460, 103109502780, 449376255840, 1963920878400, 8604858967692, 37790621078040, 166329352089096, 733551460238308

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 665

Cf. A049140, A052708.

spec := [S,{C=Prod(B,B),S=Prod(C,C),B=Union(S,C,Z)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

G.f.: RootOf(-_Z+_Z^4+_Z^2+x) - RootOf(-_Z+_Z^4+_Z^2+x)^2 - x.
Recurrence: {a(1)=0, a(2)=0, a(4)=1, a(3)=0, a(5)=4, a(6)=14, (2304-4608*n-36864*n^2+73728*n^3)*a(n)+(46368+121536*n+59904*n^2-9216*n^3)*a(n+1)+(-79800-56512*n+59520*n^2+34048*n^3)*a(n+2)+(200516*n+185964+68544*n^2+7456*n^3)*a(n+3)+(-15024*n^2-22732*n+6864-2228*n^3)*a(n+4)+(-217*n^3-2604*n^2-10199*n-13020)*a(n+5)}.
a(n) = A049140(n) - A052708(n), n>1. - R. J. Mathar, Jan 13 2025

A369215 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x) ).

Original entry on oeis.org

1, 4, 29, 261, 2627, 28315, 319648, 3731037, 44663058, 545312504, 6764556591, 85015779095, 1080185111768, 13852183882612, 179058158369828, 2330621446075640, 30519758687849439, 401806204894374041, 5315243189757111099, 70613088335938995385, 941714812929017751855

Offset: 0

Seiichi Manyama, Jan 16 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..868
Index entries for reversions of series

Cf. A369114, A369161.
Cf. A151374, A249924, A369216.
Cf. A365764, A379171, A379172.
Cf. A049140.

CoefficientList[InverseSeries[Series[x((1-x)^3-x),{x,0,21}],x]/x,x] (* Stefano Spezia, Mar 31 2025 *)

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

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

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

A217358 Series reversion of x-x^3-x^4.

Original entry on oeis.org

1, 0, 1, 1, 3, 7, 16, 45, 110, 308, 819, 2275, 6328, 17748, 50388, 143412, 411939, 1187329, 3441559, 10015005, 29255655, 85766655, 252201690, 743819115, 2199446652, 6519727800, 19369551936, 57665571072, 172011364452, 514021640564, 1538650042952

Offset: 1

R. J. Mathar, Oct 01 2012

nonn

			If y= x-x^3-x^4, then x= y + y^3 + y^4 +3*y^5 +7*y^6 +16*y^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A049140 (reversion of x-x^2-x^4).

Rest[CoefficientList[InverseSeries[Series[x - x^3 - x^4, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Sep 10 2013 *)

Conjecture: 46*n*(n-1)*(n-2)*a(n) -(n-1)*(n-2)*(11*n-74)*a(n-1) -(n-2)*(336*n^2-1359*n+1351)*a(n-2) +(-347*n^3+2190*n^2-3861*n+1330)*a(n-3) + 8*(2*n-7)*(4*n-15)*(4*n-17)*a(n-4) = 0.
Recurrence (order 3): 23*(n-2)*(n-1)*n*(9*n-25)*a(n) = -(n-2)*(n-1)*(54*n^2 - 231*n + 248)*a(n-1) + (n-2)*(1485*n^3 - 10065*n^2 + 22292*n - 16088)*a(n-2) + 8*(2*n-5)*(4*n-13)*(4*n-11)*(9*n-16)*a(n-3). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 3/23*(2367+966*sqrt(3))^(1/3)+423/(23*(2367+966*sqrt(3))^(1/3))-2/23 = 3.145200906807902443... is the root of the equation -256 - 165*d + 6*d^2 + 23*d^3 = 0 and c = 1/48*sqrt(2)*sqrt((80793 + 65184*sqrt(3))^(1/3)*((80793 + 65184 * sqrt(3))^(2/3)-1839+9*(80793 + 65184 * sqrt(3))^(1/3)))/((80793 + 65184 * sqrt(3))^(1/3)*sqrt(Pi)) = 0.098446219937815765... - Vaclav Kotesovec, Sep 10 2013

A368931 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^3) ).

Original entry on oeis.org

1, 2, 7, 31, 154, 819, 4560, 26244, 154874, 932074, 5698745, 35297535, 221016593, 1396717756, 8896798020, 57062237502, 368201804973, 2388587515239, 15568995139404, 101913055166811, 669678357109300, 4415837460391845, 29210203356645090

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A236339, A368932, A368933.

Cf. A049140, A054514.

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

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

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

A370624 Coefficient of x^n in the expansion of 1 / (1-x-x^3)^n.

Original entry on oeis.org

1, 1, 3, 13, 55, 231, 987, 4278, 18711, 82390, 364793, 1622556, 7244419, 32449158, 145747290, 656199048, 2960596359, 13382107227, 60587421882, 274712295550, 1247233045905, 5669390005950, 25798654040580, 117513750346200, 535766200488675, 2444698473079356

Offset: 0

Seiichi Manyama, May 01 2024

nonn

Cf. A049140.

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

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

A365268 G.f. satisfies A(x) = 1 + xA(x)^2(1 + x^3*A(x)^2).

Original entry on oeis.org

1, 1, 2, 5, 15, 48, 160, 549, 1929, 6909, 25134, 92612, 344924, 1296376, 4910656, 18728645, 71857133, 277160183, 1074085446, 4180057725, 16329796959, 64014638564, 251734985808, 992788252700, 3925688845948, 15560762343388, 61818928594952

Offset: 0

Seiichi Manyama, Aug 30 2023

nonn

Cf. A006605, A049140.

Cf. A063021, A365267.

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

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

A369214 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^3) ).

Original entry on oeis.org

1, 2, 7, 31, 155, 833, 4696, 27393, 163944, 1001022, 6211049, 39048685, 248213672, 1592561156, 10300192220, 67083304750, 439571860881, 2895898913453, 19169805142929, 127442939722175, 850536450459795, 5696270624620125, 38271171118343550

Offset: 0

Seiichi Manyama, Jan 16 2024

nonn

Index entries for reversions of series

Cf. A151374, A249924, A369160.

Cf. A049140, A369114.

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

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

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

A259938 Expansion of the series reversion of Sum_{n>=1} x^(n^2).

Original entry on oeis.org

0, 1, 0, 0, -1, 0, 0, 4, 0, -1, -22, 0, 13, 140, 0, -136, -970, 9, 1330, 7104, -231, -12650, -54096, 3900, 118780, 423890, -54810, -1108380, -3393696, 695640, 10311840, 27615648, -8282604, -95810606, -227480848, 94449456, 889817328, 1890685212, -1044402840, -8263944216, -15811484852

Offset: 0

Vladimir Reshetnikov, Jul 09 2015

x + x^4 + x^9 + x^16 + x^25 + ... is the expansion of (theta_3(0, x) - 1)/2, where theta_3 is the Jacobi theta function.

Max Alekseyev, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A049140, A192540.

InverseSeries[(EllipticTheta[3, 0, x] - 1)/2 + O[x]^30][[3]]

Vec( serreverse( sum(i=1,32,x^i^2) + O(x^33^2) ) ); \\ Max Alekseyev, Jul 06 2021

For n>1, a(n) = Sum_{j2,j3,...} (-1)^(j2+j3+...) * (n-1+j2+j3+...)! / (j2!*j3!*...) / n!, where the sum is taken over all nonnegative integers j2, j3, ... such that (2^2-1)*j2 + (3^2-1)*j3 + ... = n-1. - Max Alekseyev, Jul 06 2021

A383479 Number of lattice paths from (0,0) to (n,n) using steps (1,0),(3,0),(0,1).

Original entry on oeis.org

1, 2, 6, 24, 100, 420, 1792, 7752, 33858, 148940, 658944, 2929056, 13070876, 58521344, 262754040, 1182619280, 5334172518, 24104916504, 109111142376, 494630028200, 2245300152480, 10204575481320, 46429481139000, 211460450151600, 963971663881200, 4398118872144192

Offset: 0

Seiichi Manyama, Apr 28 2025

nonn

Robert Israel, Table of n, a(n) for n = 0..1492

Cf. A038112, A383480.

Cf. A049140, A144401, A370624.

f:= proc(x,y) option remember;
     local t;
     t:= 0;
     if x >= 1 then t:= t + procname(x-1,y) fi;
     if x >= 3 then t:= t + procname(x-3,y) fi;
     if y >= 1 then t:= t + procname(x,y-1) fi;
     t
end proc:
f(0,0):= 1:
seq(f(n,n),n=0..25); # Robert Israel, May 28 2025

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

a(n) = [x^n] 1/(1 - x - x^3)^(n+1).
a(n) = (n+1) * A049140(n+1).
a(n) = Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(2*n-2*k,n-3*k).

cp's OEIS Frontend

A052708 A simple context-free grammar: convolution square of A049140.

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Maple

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A052710 A simple context-free grammar: difference between A049140 and its convolution square.

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Maple

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A369215 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x) ).

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Mathematica

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PARI

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A217358 Series reversion of x-x^3-x^4.

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Mathematica

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A368931 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^3) ).

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PARI

PARI

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A370624 Coefficient of x^n in the expansion of 1 / (1-x-x^3)^n.

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PARI

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A365268 G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + x^3*A(x)^2).

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PARI

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A369214 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^3) ).

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PARI

PARI

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A259938 Expansion of the series reversion of Sum_{n>=1} x^(n^2).

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A365268 G.f. satisfies A(x) = 1 + xA(x)^2(1 + x^3*A(x)^2).