Original entry on oeis.org
10, 135, 1844, 25145, 342846, 4674655, 63738280, 869062689, 11849550290, 161566989191, 2202943686300, 30036834314425, 409548106582534, 5584132130887935, 76138873929651536, 1038143078887634945, 14154938162574828570, 193000635905606023879, 2631537137933532600580
Offset: 0
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CoefficientList[Series[(10-5x+14x^2-x^3)/(1-14x+6x^2-14x^3+x^4),{x,0,30}],x] (* Harvey P. Dale, Dec 09 2018 *)
A078817
Table by antidiagonals giving variants on Catalan sequence: T(n,k)=C(2n,n)*C(2k,k)*(2k+1)/(n+k+1).
Original entry on oeis.org
1, 3, 1, 10, 4, 2, 35, 15, 9, 5, 126, 56, 36, 24, 14, 462, 210, 140, 100, 70, 42, 1716, 792, 540, 400, 300, 216, 132, 6435, 3003, 2079, 1575, 1225, 945, 693, 429, 24310, 11440, 8008, 6160, 4900, 3920, 3080, 2288, 1430, 92378, 43758, 30888, 24024, 19404
Offset: 0
Rows start:
1, 3, 10, 35, 126, 462, 1716,
1, 4, 15, 56, 210, 792, 3003,
2, 9, 36, 140, 540, 2079, 8008,
5, 24, 100, 400, 1575, 6160, 24024,
14, 70, 300, 1225, 4900, 19404, 76440,
42, 216, 945, 3920, 15876, 63504,252252,
132, 693, 3080, 12936, 52920,213444,853776,
etc.
- Ira Gessel, Super ballot numbers, J. Symbolic Computation 14 (1992), 179-194.
- Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
- Jovan Mikic, A Note on the Gessel Numbers, arXiv:2203.12931 [math.CO], 2022.
Essentially a reflected version of
A033820.
-
A078817 := proc(n,k)
binomial(2*n,n)*binomial(2*k,k)*(2*k+1)/(n+k+1) ;
end proc: # R. J. Mathar, Dec 06 2018
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
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Table[140*Binomial[2*n, n]/(n + 4), {n, 0, 30}] (* Amiram Eldar, Feb 16 2023 *)
Showing 1-3 of 3 results.