A127167
T(3n,2n), where T is the array in A047110.
Original entry on oeis.org
1, 5, 73, 1348, 27811, 612728, 14103464, 334974405, 8146520511, 201822398131, 5074951075766, 129185072614240, 3322359273912432, 86191998671455630, 2252923797781463037, 59273686760263160137, 1568440076774389559527, 41713725234702452284249
Offset: 0
T(h,k) counts lattice paths restricted as in A047110 by the line y=2x/3, so that the numbers T(3n,2n) are of interest.
T(3,2) counts these 5 paths: RRRUU, RRURU, URRRU, URURR, UURRR.
-
T:= proc(h, k) option remember;
`if`([h, k]=[0, 0], 1, `if`(h<0 or k<0, 0, T(h-1, k)+
`if`(3*k>2*h and 3*(k-1)<2*h, 0, T(h, k-1))))
end:
a:= n-> T(3*n, 2*n):
seq (a(n), n=0..30); # Alois P. Heinz, Apr 04 2012
series(RootOf((161051*x^2+1088*x-64)*A^10 + (4600*x-320)*A^9 + (7276*x-624)*A^8 + (7830*x-512)*A^7 + (6189*x+68)*A^6 + (2177*x+540)*A^5 +535*A^4 +278*A^3 +84*A^2 +14*A +1, A), x=0, 30); # Mark van Hoeij, May 01 2013
-
T[h_, k_] := T[h, k] = If[{h, k} == {0, 0}, 1, If[h < 0 || k < 0, 0, T[h-1, k] + If[3k > 2h && 3(k-1) < 2h, 0, T[h, k-1]]]];
a[n_] := T[3n, 2n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 21 2025, after Alois P. Heinz *)
A047111
Sum{T(i,n-i): i=0,1,...,n}, array T as in A047110.
Original entry on oeis.org
1, 2, 3, 6, 10, 20, 40, 71, 142, 260, 520, 1040, 1936, 3872, 7330, 14660, 29320, 55874, 111748, 215252, 430504, 861008, 1663662, 3327324, 6476960, 12953920, 25907840, 50511295, 101022590, 198016168, 396032336, 792064672, 1553798279
Offset: 0
Original entry on oeis.org
1, 1, 2, 10, 29, 91, 370, 1305, 4626, 17888, 66824, 248839, 955688, 3646596, 13892482, 53548120, 206439059, 795884541, 3083453744, 11962794031, 46444491143, 180751142106, 704383034528, 2747337971249, 10730723024786
Offset: 0
Original entry on oeis.org
1, 2, 5, 10, 29, 164, 534, 1839, 7813, 28888, 106712, 423250, 1621048, 6175114, 24116312, 93457176, 361166949, 1408515216, 5488264319, 21368029926, 83497729588, 326481132377, 1276713094911, 5001089844661, 19605767057292
Offset: 1
Original entry on oeis.org
1, 3, 9, 24, 73, 164, 534, 3187, 11000, 39888, 174411, 665360, 2528518, 10223830, 39909056, 154727890, 612630675, 2404810575, 9405235895, 37053238445, 145729990271, 572330060383, 2253751873412, 8875044032506
Offset: 2
Original entry on oeis.org
1, 4, 14, 44, 144, 414, 1348, 3187, 11000, 67699, 242110, 907470, 4048716, 15792744, 61270714, 251463726, 996295359, 3916971576, 15685208519, 62232260683, 245848928006, 977038778501, 3873954187845, 15330413916232
Offset: 3
Original entry on oeis.org
1, 2, 5, 19, 62, 206, 771, 2787, 10075, 37936, 142127, 532438, 2025548, 7717368, 29431808, 112981883, 434717592, 1674938528, 6474529712, 25076461217, 97253412518, 377901902151, 1470624876338, 5729633180125, 22352003680730
Offset: 0
Original entry on oeis.org
1, 3, 9, 33, 115, 401, 1482, 5449, 20048, 75303, 283599, 1069860, 4070772, 15539326, 59433763, 228278533, 879053987, 3391075968, 13113667186, 50808921375, 197150760045, 766241841810, 2982295208876, 11621280655944
Offset: 0
Original entry on oeis.org
1, 4, 14, 53, 195, 711, 2662, 9973, 37367, 141472, 537422, 2045224, 7821958, 30001955, 115296650, 444336395, 1716137440, 6639137474, 25732460158, 99897347527, 388339939659, 1511670332538, 5891647475819, 22987227251180
Offset: 0
Original entry on oeis.org
1, 2, 9, 44, 250, 1418, 8567, 51635, 321039, 1996724, 12628528, 80011583, 511889348, 3282172599, 21172582733, 136879748346, 888481523846, 5778565689270, 37689416642042, 246245960185175, 1612234572834084
Offset: 0
Clark Kimberling; definition revised Dec 08 2006; sequence corrected Dec 18 2006
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