A368961 -id:A368961 - cp's OEIS frontend

A368962 Expansion of (1/x) * Series_Reversion( x * (1-x-x^3)^2 ).

1, 2, 7, 32, 165, 910, 5251, 31314, 191463, 1193808, 7561825, 48522630, 314752515, 2060587112, 13597183916, 90342651982, 603886553067, 4058197580308, 27401404029181, 185806213609730, 1264774546754103, 8639226724499070, 59198404680049915

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A368966, A368968.

Cf. A368961.

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

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

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

A368967 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^2)^2 ).

Original entry on oeis.org

1, 4, 28, 238, 2244, 22568, 237199, 2574276, 28627224, 324503718, 3735672880, 43555658640, 513277420803, 6103767231712, 73153216133600, 882708243017414, 10714917867247020, 130752597362068496, 1603069096165788706, 19737123968746454284, 243930175282166574432

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..893
Index entries for reversions of series

Cf. A368961, A368965.

Cf. A368968.

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

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(5*n-k+3,n-2*k).

A368965 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^2)^2 ).

Original entry on oeis.org

1, 3, 17, 117, 895, 7309, 62410, 550431, 4975297, 45846977, 429095387, 4067760593, 38977419018, 376901628882, 3673226867356, 36043590216621, 355800292078095, 3530878133357175, 35205183620396571, 352505713454687599, 3543078943592291301

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..971
Index entries for reversions of series

Cf. A368961, A368967.

Cf. A368966.

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

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

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

A368969 Expansion of (1/x) * Series_Reversion( x * (1-x+x^2)^2 ).

Original entry on oeis.org

1, 2, 5, 12, 22, 0, -284, -1938, -9367, -36938, -118105, -260130, 56637, 4890560, 35945616, 186674620, 782890326, 2632462236, 5987222046, -2241224328, -129137211280, -967479390360, -5145272296080, -22060975744080, -75535676951124, -172915138783080

Offset: 0

Seiichi Manyama, Jan 10 2024

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A368973, A368975.

Cf. A368961.

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

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

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

A371574 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) (1 + x*A(x)) )^2.

Original entry on oeis.org

1, 2, 13, 106, 986, 9902, 104641, 1146654, 12910674, 148462310, 1736178005, 20584835962, 246874102771, 2989580399330, 36504669373240, 448960388422126, 5556453433915920, 69150493021938224, 864833621158491876, 10863849369160145222, 137011477676531989664

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A000108, A143927, A365153, A368961, A371575.

Cf. A365186.

a(n, r=2, s=1, t=5, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

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

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A365186.

A369510 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).

Original entry on oeis.org

1, 4, 28, 240, 2288, 23296, 248064, 2728704, 30764800, 353633280, 4128783360, 48827351040, 583674642432, 7041154416640, 85610725769216, 1048040981594112, 12907157115568128, 159802897621319680, 1987875305403187200, 24833149969036738560, 311409431144819589120

Offset: 0

Seiichi Manyama, Jan 25 2024

nonn

a(n) also counts triangulations of a convex (2n+3)-gon whose points are colored red and blue alternatingly, and that do not have monochromatic triangles (i.e., every triangle has at least one red point and at least one blue point). - Torsten Muetze, May 08 2024

Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, Ars Combinatoria, 88 (2008), 109-124.

CombOS - Combinatorial Object Server, Generate k-ary trees and dissections
Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, arXiv:math/0407280 [math.CO], 2004.
Index entries for reversions of series

Cf. A368961, A369513.
Cf. A151374.
Cf. A153231 (colorful triangulations with an even number of points).

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

a(n) = sum(k=0, n\2, binomial(2*n+k+1, k)*binomial(5*n+3, n-2*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(5*n+3,n-2*k).
From Torsten Muetze, May 08 2024: (Start)
a(n) = 2^n/(n+1) * binomial(3n+1,n).
a(n) = 2^n*A006013(n). (End)

A369486 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^2)^2 ).

Original entry on oeis.org

1, 1, 4, 15, 67, 314, 1547, 7865, 41004, 217953, 1176832, 6436676, 35587416, 198569471, 1116741601, 6323669519, 36024382515, 206315985386, 1187205083042, 6860598312545, 39797882898452, 231666709974264, 1352813494962672, 7922553881534274, 46520280837291427

Offset: 0

Seiichi Manyama, Jan 24 2024

nonn

Index entries for reversions of series

Cf. A368957, A368961, A368965, A368967.

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

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

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

A370617 Coefficient of x^n in the expansion of 1 / (1-x-x^2)^(2*n).

Original entry on oeis.org

1, 2, 14, 98, 726, 5522, 42770, 335512, 2656998, 21195944, 170076214, 1371181110, 11098310730, 90128497032, 734008622872, 5992486341248, 49028047353670, 401885885751630, 3299812135410080, 27134786911366212, 223433635272820126, 1842041118321640390

Offset: 0

Seiichi Manyama, Apr 30 2024

nonn

Cf. A370618, A370619.

Cf. A368961.

a(n, s=2, t=2, 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/2)} binomial(2*n+k-1,k) * binomial(3*n-k-1,n-2*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^2)^2 ). See A368961.

A371575 G.f. satisfies A(x) = ( 1 + xA(x)^3 (1 + x*A(x)) )^2.

Original entry on oeis.org

1, 2, 15, 144, 1587, 18942, 238301, 3111788, 41779164, 573127760, 7998164674, 113189243386, 1620583793262, 23431706243230, 341654376602948, 5017986762425680, 74170837061591036, 1102479579201183898, 16469074050937364044, 247115476148847822586

Offset: 0

Seiichi Manyama, Mar 28 2024

nonn

Cf. A000108, A143927, A365153, A368961, A371574.

a(n, r=2, s=1, t=6, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

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

A376326 Expansion of (1/x) * Series_Reversion( x * (1-x-x^2)^4 ).

Original entry on oeis.org

1, 4, 30, 272, 2737, 29380, 329614, 3818540, 45329440, 548511612, 6740687924, 83898110660, 1055441468145, 13398494365088, 171422870731600, 2208161418665872, 28614197357895055, 372754395074051500, 4878709294080115494, 64123505084010848580, 846018700129069313495

Offset: 0

Seiichi Manyama, Sep 20 2024

nonn

Index entries for reversions of series

Cf. A001002, A368961, A368963.

Cf. A365188.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x-x^2)^4)/x)

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(4*n+k+3,k) * binomial(5*n-k+3,n-2*k).

G.f.: B(x)^4, where B(x) is the g.f. of A365188.

cp's OEIS Frontend

A368962 Expansion of (1/x) * Series_Reversion( x * (1-x-x^3)^2 ).

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A368967 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^2)^2 ).

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A368965 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^2)^2 ).

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A368969 Expansion of (1/x) * Series_Reversion( x * (1-x+x^2)^2 ).

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A371574 G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) * (1 + x*A(x)) )^2.

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A369510 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^2)^2 ).

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A369486 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^2)^2 ).

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A370617 Coefficient of x^n in the expansion of 1 / (1-x-x^2)^(2*n).

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A371575 G.f. satisfies A(x) = ( 1 + x*A(x)^3 * (1 + x*A(x)) )^2.

A371574 G.f. satisfies A(x) = ( 1 + xA(x)^(5/2) (1 + x*A(x)) )^2.

A371575 G.f. satisfies A(x) = ( 1 + xA(x)^3 (1 + x*A(x)) )^2.