A046736 -id:A046736 - cp's OEIS frontend

A049125 Revert transform of (1 + x - x^2) / (1 + x)^2.

1, 1, 2, 4, 10, 25, 68, 187, 534, 1544, 4554, 13576, 40968, 124681, 382636, 1182116, 3674674, 11483243, 36057516, 113701968, 359927638, 1143327888, 3643379152, 11643793399, 37311200060, 119852247220, 385864664018, 1244896820476

Offset: 1

Olivier Gérard

nonn

a(n) is the number of ordered trees (A000108) with n edges in which every non-leaf non-root vertex has at most one leaf child. The g.f. A(x) is given by A(x)= x/(1-x B(x)) where B(x)=1+x+2x^2+4x^3+... is the g.f. for A143363. [David Callan, Aug 22 2014]
Conjecturally, the number of sequences (e(1), ..., e(n-1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(k). [Martinez and Savage, 2.10] - Eric M. Schmidt, Jul 17 2017
a(n) is the number of dissections of a convex (n+m)-sided polygon by non-intersecting diagonals such that the selected m consecutive sides of the polygon will be in the same subpolygon and  create no triangles. - Muhammed Sefa Saydam, Jul 12 2025

			x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 25*x^6 + 68*x^7 + 187*x^8 + 534*x^9 + ...

Wenqin Cao, Emma Yu Jin, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Index entries for reversions of series

a[1] = 1;
a[n_] := SeriesCoefficient[InverseSeries[x(1+x-x^2)/(1+x)^2 + x O[x]^n, x], {x, 0, n}];
Array[a, 28] (* Jean-François Alcover, Aug 17 2018, from PARI *)
CoefficientList[InverseSeries[Series[x*(1 + x - x^2)/(1 + x)^2, {x, 0, 30}], x], x] (* Vaclav Kotesovec, Aug 17 2018 *)

{a(n) = if( n<1, 0, polcoeff( serreverse( x * (1 + x - x^2) / (1 + x)^2 + x * O(x^n)), n))} /* Michael Somos, Jul 13 2003 */

Given g.f. A(x), then series reversion of B(x) = x + x * A(x) is -B(-x). - Michael Somos, Sep 07 2005
Given g.f. A(x), then B(x) = x + x * A(x) satisfies B(x) = x + C(x * B(x)) where C(x) is g.f. of A001764 offset 1.
D-finite with recurrence 5*n*(n-1)*(37*n-106)*a(n) -4*(n-1) *(74*n^2-323*n+288)*a(n-1) +16*(-74*n^3+508*n^2-1157*n+876)* a(n-2) +2*(2*n-5)*(37*n-69)*(n-4)*a(n-3)=0. - R. J. Mathar, Jun 24 2018
a(n) ~ (1+s)^2 / (2 * sqrt(Pi*(1+4*s)) * n^(3/2) * (s*(1 + s - s^2)/(1+s)^2)^(n - 1/2)), where s = 0.675130870566646070889621798150060480808032527677372732 = 2*cos(arctan(sqrt(37/27))/3)/sqrt(3) + 2*sin(arctan(sqrt(37/27))/3) - 1 is the root of the equation s^3 + 3*s^2 - s = 1. - Vaclav Kotesovec, Aug 17 2018
From Muhammed Sefa Saydam, Jul 12 2025: (Start)
a(n) = Sum_{k=1..n+1} A046736(k) * A046736(n-k+2), for A046736(1) = 1 and n >= 2.
a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * A046736(n-k+1), for a(0) = 1 and n >= 3.
a(n) = a(n-1) + Sum_{k=1..n-2} A143363(k) * a(n-k-1), for a(0) = 1 and n >= 2. (End)

A253194 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no pentagons.

Original entry on oeis.org

1, 3, 10, 39, 162, 707, 3190, 14766, 69719, 334481, 1625846, 7989908, 39631204, 198151579, 997623275, 5053274850, 25734158411, 131680565544, 676693557574, 3490897656337, 18071699948492, 93851485181749, 488815126122166

Offset: 1

Michael D. Weiner, Mar 24 2015

nonn

			a(3)=10 because the pentagon allows all but the null placement, i.e., 5 placements of 1 diagonal and 5 placements of two diagonals.

D. Birmajer, J. B. Gil, and M. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv:1503.05242 [math.CO], 2015.

Cf. A046736, A054515.

Rest[CoefficientList[(InverseSeries[Series[(y-2*y^2+y^4-y^5)/(1-y),{y,0,24}],x]-x)/x,x]]

A253194(n)=sum(i=0,(n-1)\3,sum(k=i+1,n-2*i, (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1)),if(n%3==0,(-1)^(n/3)*binomial(4*n/3,n/3)))/(n+1) \\ M. F. Hasler, Apr 07 2015

a(n) = (1/(n+1))*Sum_{i=0..floor(n/3)} Sum_{k=i+1..n-2*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1), n !== 0 (mod 3),
a(n) = ((-1)^(n/3)/(n+1))*binomial(4*n/3,n/3) + (1/(n+1))*Sum_{i=0..(n/3)-1} Sum_{k=i+1..n-2*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-3*i-1,k-i-1), n == 0 (mod 3).
Recurrence: 275*(n-2)*(n-1)*n*(n+1)*(13962464*n^5 - 196202616*n^4 + 1069508732*n^3 - 2802358002*n^2 + 3480787751*n - 1597000860)*a(n) = 900*(n-2)*(n-1)*n*(27924928*n^6 - 406367696*n^5 + 2336399896*n^4 - 6678345644*n^3 + 9735406192*n^2 - 6526643891*n + 1424056473)*a(n-1) - 8*(n-2)*(n-1)*(1870970176*n^7 - 30033090896*n^6 + 197840216728*n^5 - 682911269612*n^4 + 1297104157966*n^3 - 1273486799084*n^2 + 486871358313*n + 21712608900)*a(n-2) - 16*(n-2)*(2569093376*n^8 - 47662201536*n^7 + 375012676176*n^6 - 1627682459628*n^5 + 4239503473896*n^4 - 6734585440155*n^3 + 6299789310412*n^2 - 3112752665481*n + 598681926090)*a(n-3) + 16*(2457393664*n^9 - 54190809728*n^8 + 518749193184*n^7 - 2816319789288*n^6 + 9492888047100*n^5 - 20388222826734*n^4 + 27407291375141*n^3 - 21462176121217*n^2 + 8117426803296*n - 745665750648)*a(n-4) - 8*(n-4)*(2*n - 7)*(4*n - 17)*(4*n - 11)*(13962464*n^5 - 126390296*n^4 + 424322908*n^3 - 631422862*n^2 + 369599799*n - 31302531)*a(n-5). - Vaclav Kotesovec, Mar 30 2015

A365691 G.f. satisfies A(x) = 1 + x^2A(x)^5 / (1 - xA(x)).

Original entry on oeis.org

1, 0, 1, 1, 6, 12, 54, 147, 593, 1886, 7292, 25204, 96153, 348304, 1327716, 4946471, 18936366, 71827598, 276612103, 1062220253, 4115807184, 15947902376, 62148513732, 242485933208, 949828266722, 3726623622402, 14663689944397, 57798199213989

Offset: 0

Seiichi Manyama, Sep 16 2023

nonn

Cf. A004148, A005043, A025246, A046736, A365690.

Cf. A365693.

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

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

A366112 Expansion of (1/x) * Series_Reversion( x*(1-x-x^5)/(1-x) ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 7, 14, 22, 31, 41, 103, 235, 457, 791, 1261, 2399, 5015, 10257, 19676, 35296, 65170, 127520, 256187, 507601, 969495, 1834433, 3534477, 6962249, 13809538, 27061252, 52439361, 101701035, 199152071, 393332277, 776589611, 1525416837

Offset: 0

Seiichi Manyama, Sep 29 2023

nonn

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

Cf. A046736, A054514, A215342.

Cf. A017899.

CoefficientList[InverseSeries[Series[x*(1-x-x^5)/(1-x),{x,0,41}]]/x,x] (* Stefano Spezia, Aug 21 2025 *)

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

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

A368934 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^2) ).

Original entry on oeis.org

1, 3, 16, 104, 751, 5789, 46656, 388377, 3313304, 28816513, 254548840, 2277498340, 20596833817, 187974816142, 1729033498416, 16012809644088, 149186508912927, 1397300099214753, 13149137686976324, 124262625068365924, 1178796712807563025

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A001002, A046736, A236339.

Cf. A368938.

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

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

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

A090985 Triangle read by rows: T(n,k) is the number of dissections of a convex n-gon by nonintersecting diagonals, having exactly k triangles (n >= 2, k >= 0).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 5, 0, 5, 4, 6, 21, 0, 14, 8, 35, 28, 84, 0, 42, 25, 80, 216, 120, 330, 0, 132, 64, 309, 540, 1155, 495, 1287, 0, 429, 191, 890, 2475, 3080, 5720, 2002, 5005, 0, 1430, 540, 3058, 7788, 16302, 16016, 27027, 8008, 19448, 0, 4862, 1616, 9580, 30108, 54964, 96005, 78624, 123760, 31824, 75582, 0, 16796

Offset: 2

Emeric Deutsch, Feb 28 2004

T(n,n-2) = [binomial(2n-4, n-2)]/(n-1) = Catalan(n-2) (A000108).
T(n,n-4) = binomial(2n-5, n-4) (A002054).
T(n,n-5) = binomial(2n-6, n-5) (A002694).
T(n,0) = A046736(n).
Row sums give the little Schroeder numbers (A001003).

			T(5,1)=5 because the dissections of a convex pentagon having exactly one triangle are obtained by the placement of a diagonal between any pair of non-adjacent vertices.
T(6,0)=4 because the dissections of a convex hexagon with no triangles are obtained by the null placement and by placing one diagonal between any of the 3 pairs of opposite vertices.
Triangle starts:
  1;
  0,  1;
  1,  0,  2;
  1,  5,  0,  5;
  4,  6, 21,  0, 14;
  8, 35, 28, 84,  0, 42;
  ...

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

Cf. A000108, A002054, A002694, A046736.

T := (n,k)->binomial(n+k-2,k)*sum(binomial(n-2+k+i,i)*binomial(n-3-k-i,i-1), i=0..floor((n-2-k)/2))/(n-1): seq(seq(T(n,k),k=0..n-2),n=2..14);

T [n_, k_] := Binomial[n+k-2, k] Sum[Binomial[n-2+k+i, i] Binomial[n-3-k-i, i-1], {i, 0, (n-2-k)/2}]/(n-1);
Table[T[n, k], {n, 2, 12}, {k, 0, n-2}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)

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

G.f.: G=G(t, z) satisfies (1-t)G^3 + (1+t)zG^2 - z^2*(1+z)G + z^4 = 0.

A256752 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no hexagons.

Original entry on oeis.org

1, 3, 11, 44, 190, 859, 4015, 19248, 94117, 467575, 2353443, 11975568, 61505088, 318406927, 1659801852, 8704865907, 45898065978, 243163198928, 1293769867676, 6910165762943, 37036898772008, 199140325574519, 1073849938338566

Offset: 1

Michael D. Weiner, Apr 09 2015

nonn

			a(3)=11 because all 11 dissections of the pentagon are allowed, i.e., the null placement, 5 placements of 1 diagonal and 5 placements of two diagonals.

D. Birmajer, J. B. Gil, and M. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv:1503.05242 [math.CO], 2015.

Cf. A046736, A054515, A253194.

Rest[CoefficientList[(InverseSeries[Series[(y-2*y^2+y^5-y^6)/(1-y), {y, 0, 24}], x]-x)/x, x]]

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

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

cp's OEIS Frontend

A049125 Revert transform of (1 + x - x^2) / (1 + x)^2.

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A253194 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no pentagons.

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A365691 G.f. satisfies A(x) = 1 + x^2*A(x)^5 / (1 - x*A(x)).

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A366112 Expansion of (1/x) * Series_Reversion( x*(1-x-x^5)/(1-x) ).

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A368934 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x-x^2) ).

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A090985 Triangle read by rows: T(n,k) is the number of dissections of a convex n-gon by nonintersecting diagonals, having exactly k triangles (n >= 2, k >= 0).

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A256752 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no hexagons.

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A365691 G.f. satisfies A(x) = 1 + x^2A(x)^5 / (1 - xA(x)).