A078817 -id:A078817 - cp's OEIS frontend

A078818 a(n) = 30*binomial(2n,n)/(n+3).

10, 15, 36, 100, 300, 945, 3080, 10296, 35100, 121550, 426360, 1511640, 5408312, 19501125, 70794000, 258529200, 949074300, 3500409330, 12964479000, 48198087000, 179799820200, 672822343050, 2524918756464, 9500112378000, 35830670759000, 135439935469020

Offset: 0

Henry Bottomley, Dec 07 2002

			a(5) = 30*binomial(10,5)/8 = 945.

Muniru A Asiru, Table of n, a(n) for n = 0..300

Cf. A000108, A000984, A001622, A007946, A038629, A078817, A078819.

List([0..30],n->30*Binomial(2*n,n)/(n+3)); # Muniru A Asiru, Aug 09 2018

[30*Binomial(2*n,n)/(n+3): n in [0..30]]; // Vincenzo Librandi, Aug 11 2018

Table[(30 Binomial[2 n, n] / (n + 3)), {n, 0, 30}] (* Vincenzo Librandi, Aug 11 2018 *)

D-finite with recurrence a(n) = a(n-1)*(4n^2+6n-4)/(n^2+3n) = A078817(n, 2) = 5*A007946(n)/(2n+1) = 30*A000984(n)/(n+3).
From Amiram Eldar, Feb 16 2023: (Start)
Sum_{n>=0} 1/a(n) = 4*Pi/(135*sqrt(3)) + 7/45.
Sum_{n>=0} (-1)^n/a(n) = 9/125 - 32*log(phi)/(375*sqrt(5)), where phi is the golden ratio (A001622). (End)

A078819 a(n) = 140*C(2n,n)/(n+4).

Original entry on oeis.org

35, 56, 140, 400, 1225, 3920, 12936, 43680, 150150, 523600, 1847560, 6584032, 23661365, 85652000, 312018000, 1142971200, 4207562730, 15557374800, 57750861000, 215145084000, 804104751450, 3014244096864, 11329763650800, 42691863032000, 161238018415500, 610258100044320

Offset: 0

Henry Bottomley, Dec 07 2002

			a(5)=140*C(10,5)/9=3920

Cf. A000108, A000984, A001622, A038629, A078817, A078818, A078820.

Table[140*Binomial[2*n, n]/(n + 4), {n, 0, 30}] (* Amiram Eldar, Feb 16 2023 *)

D-finite with recurrence a(n) = a(n-1)*(4n^2+10n-6)/(n^2+4n) = A078817(n, 3) = 7*A078820(n)/(2n+1) = 140*A000984(n)/(n+4).
From Amiram Eldar, Feb 16 2023: (Start)
Sum_{n>=0} 1/a(n) = Pi/(126*sqrt(3)) + 3/70.
Sum_{n>=0} (-1)^n/a(n) = 37/1750 - 3*log(phi)/(125*sqrt(5)), where phi is the golden ratio (A001622). (End)

A078820 a(n) = 20C(2n,n)(2n+1)/(n+4).

Original entry on oeis.org

5, 24, 100, 400, 1575, 6160, 24024, 93600, 364650, 1421200, 5542680, 21633248, 84504875, 330372000, 1292646000, 5061729600, 19835652870, 77786874000, 305254551000, 1198665468000, 4709756401350, 18516070880736, 72834194898000, 286645366072000, 1128666128908500

Offset: 0

Henry Bottomley, Dec 07 2002

			a(5)=20*C(10,5)*11/9=6160.

Cf. A001622, A001700, A001791, A002457, A007946, A078817, A078819.

Table[20Binomial[2n,n] (2n+1)/(n+4),{n,0,30}] (* Harvey P. Dale, Nov 02 2011 *)

D-finite with recurrence a(n) = a(n-1)*(4n^2+14n+6)/(n^2+4n) = A078817(3, n) = (2n+1)*A078819(n)/7 = 20*A002457(n)/(n+4).
From Amiram Eldar, Feb 16 2023: (Start)
Sum_{n>=0} 1/a(n) = 11*Pi/(90*sqrt(3)) + 1/30.
Sum_{n>=0} (-1)^n/a(n) = 17*log(phi)/(25*sqrt(5)) + 1/50, where phi is the golden ratio (A001622). (End)

A033820 Triangle read by rows: T(k,j) = ((2j+1)/(k+1))binomial(2j,j)binomial(2k-2j,k-j).

Original entry on oeis.org

1, 1, 3, 2, 4, 10, 5, 9, 15, 35, 14, 24, 36, 56, 126, 42, 70, 100, 140, 210, 462, 132, 216, 300, 400, 540, 792, 1716, 429, 693, 945, 1225, 1575, 2079, 3003, 6435, 1430, 2288, 3080, 3920, 4900, 6160, 8008, 11440, 24310, 4862, 7722, 10296, 12936, 15876, 19404

Offset: 0

Alexander Burstein

f(n,k) = 2^{n-2(k-2)}sum(T(k-2,j)*binomial(n+2*(k-2-j),2*(k-2-j)),j=0..k-2) is the number of length n k-ary strings (k >= 2) which avoid a rising triple (pattern 123) or any other given 3-letter permutation pattern.

Row sums are the powers of 4. This is explained by a simple statistic on the 4^n lattice paths of length 2n formed from upsteps U=(1,1) and downsteps D=(1,-1). For such a path, define X = number of upsteps that lie above ground level (GL), the horizontal line through the initial vertex, and before the last vertex at GL. For UDDUUUUDDU for instance, the last vertex at GL follows the fourth step, and so X = 1. T(n,k) is the number of these paths with X=n-k. For example, T(2,1)=4 counts UDUU, UDDU, UDDD, DUUD because each has n-k=1 upsteps above GL and before the last vertex at GL. - David Callan, Nov 21 2011

			{1},
{1, 3},
{2, 4, 10},
{5, 9, 15, 35},
{14, 24, 36, 56, 126},
{42, 70, 100, 140, 210, 462},
{132, 216, 300, 400, 540, 792, 1716},
...

Alexander Burstein, Enumeration of words with forbidden patterns, Ph.D. thesis, University of Pennsylvania, 1998.
Ira Gessel, Super ballot numbers, J. Symbolic Computation 14 (1992), 179-194.
Walter Shur, Two Game-Set Inequalities, J. Integer Seqs., Vol. 6, 2003.

Cf. A000108, A000984, A000302.

Essentially a reflected version of A078817.

T(k,0) = binomial(2*k, k)/(k+1), the k-th Catalan number; T(k,k) = binomial(2*(k+1),k+1)/2, half the (k+1)-st central binomial coefficient sum of entries in row k (over j) = 2^{2*(k-1)}
T(k,j) = sum(C(k-i)D(i), i=0..j), C(i) = binomial(2*i,i)/(i+1), D(i) = binomial(2*i,i).
G.f.: 2/(1-4*x*y+sqrt((1-4*x)*(1-4*x*y))). - Vladeta Jovovic, Dec 14 2003

More terms from Vladeta Jovovic, Dec 10 2003

cp's OEIS Frontend

A078818 a(n) = 30*binomial(2n,n)/(n+3).

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A078819 a(n) = 140*C(2n,n)/(n+4).

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A078820 a(n) = 20*C(2n,n)*(2n+1)/(n+4).

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A033820 Triangle read by rows: T(k,j) = ((2*j+1)/(k+1))*binomial(2*j,j)*binomial(2*k-2*j,k-j).

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A078820 a(n) = 20C(2n,n)(2n+1)/(n+4).

A033820 Triangle read by rows: T(k,j) = ((2j+1)/(k+1))binomial(2j,j)binomial(2k-2j,k-j).