A079489 -id:A079489 - cp's OEIS frontend

A381753 Expansion of exp( Sum_{k>=1} binomial(5k-1,2k-1) * x^k/k ).

1, 4, 50, 846, 16495, 349240, 7803823, 181135830, 4324897697, 105543188190, 2620784850325, 66005699547352, 1682046970846570, 43291586055360034, 1123707191010320955, 29382536610737191930, 773229801368332554273, 20463493681189771623960

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381752, A381757, A381758.

Cf. A060941.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(5*k-1, 2*k-1)*x^k/k)))

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

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(5*k-1,2*k-1) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A060941.
a(n) = 2 * Sum_{k=0..n} binomial(5*n+2*k+2,k) * binomial(5*n+2,n-k)/(5*n+2*k+2).

A381757 Expansion of exp( Sum_{k>=1} binomial(7k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 6, 161, 6062, 265868, 12720904, 643915209, 33905228350, 1838102210977, 101910583801012, 5751779249830131, 329359930638541776, 19087504000780665541, 1117418973753045781944, 65982722733895652916539, 3925378032146863676341770, 235048328495265879957413946

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381752, A381753, A381758.

Cf. A300386.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(7*k-1, 2*k-1)*x^k/k)))

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

G.f.: B(x)^2, where B(x) is the g.f. of A300386.

A381758 Expansion of exp( Sum_{k>=1} binomial(9k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 8, 372, 24732, 1925394, 163883548, 14773987638, 1386341339430, 133994232166575, 13248555929274096, 1333732204895318366, 136243562694021684648, 14087033746990654649067, 1471456489458490198994856, 155042502964505871862313879, 16459391575059417875255359878

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381752, A381753, A381757.

Cf. A300387.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(9*k-1, 2*k-1)*x^k/k)))

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(9*k-1,2*k-1) * a(n-k).

G.f.: B(x)^2, where B(x) is the g.f. of A300387.

A381744 Expansion of exp( Sum_{k>=1} binomial(6k-1,2k) * x^k/k ).

Original entry on oeis.org

1, 10, 215, 5942, 186111, 6283192, 222992692, 8201608382, 309834609743, 11950890428170, 468707758663887, 18634632264615272, 749325132218313540, 30422303269317412048, 1245346665979469486376, 51343805279989437688334, 2130090659402456357279919, 88858984785475871013971710

Offset: 0

Seiichi Manyama, Mar 05 2025

Seiichi Manyama, Table of n, a(n) for n = 0..605

Cf. A079489, A381745, A381746.

Cf. A006013, A182960.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(6*k-1, 2*k)*x^k/k)))

G.f. A(x) satisfies A(x^2) = B(x) * B(-x), where B(x) is the g.f. of A006013.
a(n) = Sum_{k=0..2*n} (-1)^k * A006013(k) * A006013(2*n-k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(6*k-1,2*k) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A182960.

A078391 Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 2, 2, 5, 14, 5, 4, 5, 14, 42, 14, 10, 10, 14, 42, 132, 42, 28, 25, 28, 42, 132, 429, 132, 84, 70, 70, 84, 132, 429, 1430, 429, 264, 210, 196, 210, 264, 429, 1430, 4862, 1430, 858, 660, 588, 588, 660, 858, 1430, 4862, 16796, 4862, 2860, 2145, 1848

Offset: 0

Henry Bottomley, Dec 24 2002

T(n,k) is the number of Dyck paths of semilength n+1 whose first return point to the axis have abscissa 2k+2. - Emeric Deutsch, Mar 01 2004
With offset = 1, T(n,k) is the number of binary trees with n internal nodes that have exactly k internal nodes in the left subtree, n>=1, 0<=k<=n-1. - Geoffrey Critzer, Feb 24 2013
T(n-1,k) is also the number of tilings of a triangular shape T_n (row k has length k for k=1, 2, ..., n) with n rectangular tiles (including squares) with contain a rectangular tile (n-k,k+1) for k = 0, 1, ... ,n-1, n >= 1. Let the number of tilings of T_n with n rectangular tiles (including squares) be A(n) and take A(0) = 1. Decompose these n-tilings of T_n into n disjoint and exhaustive classes C(n, k), for k = 0, 1, ..., n-1, n >= 1. In class C(n, k) one takes a fixed rectangular tile (n-k,k+1) leaving triangles T_(n-1-k) and T_k to be tiled (but for the k=0 class T_0 is not shown). Then A(n) = A(n-1)*A(0) + A(n-2)*A(1) + ... + A(0)*A(n-1) = sum(A(n-1-k)*A(k), k=0..n-1), n >= 1, with A(0)=1. But this is the recurrence for the Catalan numbers, hence A(n) = C(n). See the link with examples n = 1..7. - Wolfdieter Lang and Kival Ngaokrajang, Dec 27 2014
T(n,k) is the number of triangulations of an (n+3)-polygon using a (0,1,k+2)-triangle. - Yuchun Ji, Jan 21 2021
Alternating sum of even index 2n is A079489(n). - F. Chapoton, Aug 26 2024

			The triangle T(n,k) begins:
n\k     0    1    2    3    4    5    6    7    8    9    10 ...
0:      1
1:      1    1
2:      2    1    2
3:      5    2    2    5
4:     14    5    4    5   14
5:     42   14   10   10   14   42
6:    132   42   28   25   28   42  132
7:    429  132   84   70   70   84  132  429
8:   1430  429  264  210  196  210  264  429 1430
9:   4862 1430  858  660  588  588  660  858 1430 4862
10: 16796 4862 2860 2145 1848 1764 1848 2145 2860 4862 16796
...  Reformatted - _Wolfdieter Lang_, Dec 27 2014
----------------------------------------------------------------

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, first edition, page 225.

FindStat - Combinatorial Statistic Finder, The position of the first return of a Dyck path., The size of the left subtree.
Kival Ngaokrajang, Illustration of C(n,k), for n = 1..7, k = 0..n-1
Richard P. Stanley, Catalan Addendum (k8)

Row sums are Catalan numbers A000108(n+1). T(2*k,k) = A001246(k), k >= 0. T(n,0) = T(n,n) = A000108(n).

Cf. A067804, A079489.

nn=10;r=(1-(1-4x)^(1/2))/(2x);l=(1-(1-4x y)^(1/2))/(2x y);f[list_]:=Select[list,#>0&];Map[f,Drop[CoefficientList[Series[1+x l r,{x,0,nn}],{x,y}],1]]//Grid (* Geoffrey Critzer, Feb 24 2013 *)
Table[CatalanNumber[k]CatalanNumber[n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Nov 14 2019 *)

G.f.: C(z)*C(tz), where C(z) = (1-sqrt(1-4z))/(2z) is the g.f. of the Catalan numbers (A000108). - Emeric Deutsch, Mar 01 2004
T(n,k) = A118921(n+1,k+1)/(2*(n-k+1)). - Philippe Deléham, Dec 13 2006
When viewed as a square array, for n>0 and k>0, A(n,k) = Sum_{i=0..n-1,j=0..k-1} A[i,j]*A[n-i,k-j]. - Gerald McGarvey, Dec 30 2007

A078990 Triangle arising from (4,2) tennis ball problem, read by rows.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 10, 16, 22, 1, 6, 21, 52, 105, 158, 211, 1, 8, 36, 116, 301, 644, 1198, 1752, 2306, 1, 10, 55, 216, 678, 1784, 4088, 8144, 14506, 20868, 27230, 1, 12, 78, 360, 1320, 4064, 10896, 25872, 55354, 105704, 183284, 260864, 338444, 1, 14, 105

Offset: 0

N. J. A. Sloane, Jan 20 2003

Length of row n = 2n+1. Rows have been reversed.

			Triangle starts:
1;
1, 2,  3;
1, 4, 10, 16,  22;
1, 6, 21, 52, 105, 158, 211;
...

D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table A.1).

Final diagonal gives A079489. Row sums give A066357(n+1).

T(n,k)=if(k<0 || k>2*n,0,if(n<1,k==0,sum(j=0,k,(j+1)*T(n-1,k-j))))

A259557 a(n) = binomial(4n-1, 2n).

Original entry on oeis.org

1, 3, 35, 462, 6435, 92378, 1352078, 20058300, 300540195, 4537567650, 68923264410, 1052049481860, 16123801841550, 247959266474052, 3824345300380220, 59132290782430712, 916312070471295267, 14226520737620288370

Offset: 0

Vladimir Kruchinin, Jun 30 2015

Essentially the same as A100033.

V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Cf. A001700, A079489, A100033.

[Binomial(4*n-1, 2*n): n in [0..20]]; // Vincenzo Librandi, Jul 01 2015

Table[Binomial[4 n - 1, 2 n], {n, 0, 30}] (* Vincenzo Librandi, Jul 01 2015 *)

vector(20, n, n--; binomial(4*n-1, 2*n)) \\ Michel Marcus, Jul 01 2015

G.f. A(x)=1+x*B(x)'/B(x), where B(x) is g.f. of A079489.
a(n) = A100033(n-1) for n>0.
D-finite with recurrence n*(2*n-1)*a(n) -2*(4*n-1)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Jul 06 2015
a(n) = [x^(2*n)] 1/(1 - x)^(2*n). - Ilya Gutkovskiy, Oct 10 2017
From Peter Bala, Jun 11 2023: (Start)
a(n) = (1/2) * [x^n] ( (1 + x)^2/( 1 - x) )^(2*n) for n >= 1.
Right-hand side of the identity (1/2)*Sum_{k = 0..n} binomial(4*n,k)*binomial(3*n-k-1,n-k) = binomial(4*n-1,2*n) for n >= 1.
a(n) = [x^n] E(x)^n, where E(x) = exp( Sum_{k >= 1} A119259(k)*x^k/k ). (End)
a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(-n, k)*binomial(-3*n-k, 2*n-k) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k-1, k)*binomial(5*n-1, 2*n-k). - Peter Bala, Jun 08 2024

A349648 Expansion of g.f.: Catalan(x)/Catalan(-x).

Original entry on oeis.org

1, 2, 2, 8, 14, 64, 132, 640, 1430, 7168, 16796, 86016, 208012, 1081344, 2674440, 14057472, 35357670, 187432960, 477638700, 2549088256, 6564120420, 35223764992, 91482563640, 493132709888, 1289904147324, 6979724509184, 18367353072152, 99710350131200

Offset: 0

Alexander Burstein, Nov 23 2021

Cf. A000108, A001622, A048990 (bijection), A052707 (bijection), A006318, A079489, A246062, A333564.

gf:= (c-> c(x)/c(-x))(x-> hypergeom([1/2, 1], [2], 4*x)):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 23 2021

CoefficientList[Series[(1-Sqrt[1-4x])/(Sqrt[1+4x]-1),{x,0,24}],x]

a(2*n) = A048990(n) = A000108(2*n), n>=0.
a(2*n+1) = A052707(n+1) = 2^(2*n+1)*A000108(n), n>=0.
G.f.: A(x) = C(x)/C(-x) = (1 - sqrt(1 - 4*x))/(sqrt(1 + 4*x) - 1), where C(x) is the g.f. of A000108.
G.f.: A(x) = F(x^2) + 2*x*F(x^2)^2 = (C(x) + C(-x))/2 + 2*x*C(4*x^2), where F(x) is the g.f. of A048990.
G.f.: A(-x) = 1/A(x).
G.f.: A(x) = R(x*C(-x)^2) = 1/R(-x*C(x)^2), where R(x) is the g.f. of A006318.
G.f.: A(x) = (1 + x*C(x)*C(-x))/(1 - x*C(x)*C(-x)), see A079489 for the expansion of C(x)*C(-x).
D-finite with recurrence n*(n-1)*(n+1)*a(n) -4*(n-1)*(8*n^2-32*n+35)*a(n-2) +64*(2*n-5)*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Mar 06 2022
Sum_{n>=0} 1/a(n) = 28/15 + 2*Pi/(9*sqrt(3)) + 64*arcsin(1/4)/(75*sqrt(15)) - 12*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Apr 20 2023
G.f.: A(x) = exp( Sum_{n >= 1} binomial(4*n-2,2*n-1)*x^(2*n-1)/(2*n-1) ). - Peter Bala, Apr 28 2023

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A381753 Expansion of exp( Sum_{k>=1} binomial(5*k-1,2*k-1) * x^k/k ).

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A381757 Expansion of exp( Sum_{k>=1} binomial(7*k-1,2*k-1) * x^k/k ).

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A381758 Expansion of exp( Sum_{k>=1} binomial(9*k-1,2*k-1) * x^k/k ).

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A381744 Expansion of exp( Sum_{k>=1} binomial(6*k-1,2*k) * x^k/k ).

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A078391 Triangle read by rows: T(n,k) = Catalan(k)*Catalan(n-k).

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A078990 Triangle arising from (4,2) tennis ball problem, read by rows.

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A259557 a(n) = binomial(4*n-1, 2*n).

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A349648 Expansion of g.f.: Catalan(x)/Catalan(-x).

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A381753 Expansion of exp( Sum_{k>=1} binomial(5k-1,2k-1) * x^k/k ).

A381757 Expansion of exp( Sum_{k>=1} binomial(7k-1,2k-1) * x^k/k ).

A381758 Expansion of exp( Sum_{k>=1} binomial(9k-1,2k-1) * x^k/k ).

A381744 Expansion of exp( Sum_{k>=1} binomial(6k-1,2k) * x^k/k ).

A259557 a(n) = binomial(4n-1, 2n).