Displaying 1-10 of 24 results found.
Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).
(Formerly M1659)
+10
294
1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090
Triangle of Narayana ( A001263) with 0 <= k <= n, read by rows.
+10
41
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0, 1, 55, 825, 4950, 13860
Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.
+10
38
1, 1, 1, 1, 3, 2, 1, 6, 10, 5, 1, 10, 30, 35, 14, 1, 15, 70, 140, 126, 42, 1, 21, 140, 420, 630, 462, 132, 1, 28, 252, 1050, 2310, 2772, 1716, 429, 1, 36, 420, 2310, 6930, 12012, 12012, 6435, 1430, 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862
Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
+10
36
1, 1, 2, 1, 5, 5, 1, 9, 21, 14, 1, 14, 56, 84, 42, 1, 20, 120, 300, 330, 132, 1, 27, 225, 825, 1485, 1287, 429, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796
a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*2^i*3^(n-i), a(0)=1.
+10
24
1, 3, 15, 93, 645, 4791, 37275, 299865, 2474025, 20819307, 178003815, 1541918901, 13503125805, 119352115551, 1063366539315, 9539785668657, 86104685123025, 781343125570515, 7124072211203775, 65233526296899981, 599633539433039445, 5531156299278726663
Triangle read by rows: T(n,k) is number of Motzkin paths of length n and having k horizontal steps.
+10
22
1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 2, 0, 6, 0, 1, 0, 10, 0, 10, 0, 1, 5, 0, 30, 0, 15, 0, 1, 0, 35, 0, 70, 0, 21, 0, 1, 14, 0, 140, 0, 140, 0, 28, 0, 1, 0, 126, 0, 420, 0, 252, 0, 36, 0, 1, 42, 0, 630, 0, 1050, 0, 420, 0, 45, 0, 1, 0, 462, 0, 2310, 0, 2310, 0, 660, 0, 55, 0, 1, 132, 0, 2772, 0
a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*3^i*4^(n-i), a(0)=1.
+10
15
1, 4, 28, 244, 2380, 24868, 272188, 3080596, 35758828, 423373636, 5092965724, 62071299892, 764811509644, 9511373563492, 119231457692284, 1505021128450516, 19112961439180588, 244028820862442116, 3130592301487969948, 40333745806536135028, 521655330655122923980
Expansion of (1-3*x-sqrt(9*x^2-10*x+1))/(2*x).
+10
13
1, 4, 20, 116, 740, 5028, 35700, 261780, 1967300, 15072836, 117297620, 924612532, 7367204260, 59240277988, 480118631220, 3917880562644, 32163325863300, 265446382860420, 2201136740855700, 18329850024033012, 153225552507991140
Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.
+10
12
1, 2, 1, 6, 6, 1, 20, 30, 12, 1, 70, 140, 90, 20, 1, 252, 630, 560, 210, 30, 1, 924, 2772, 3150, 1680, 420, 42, 1, 3432, 12012, 16632, 11550, 4200, 756, 56, 1, 12870, 51480, 84084, 72072, 34650, 9240, 1260, 72, 1, 48620, 218790, 411840, 420420, 252252
a(n) = (3*n)! / ((n+1)*(n!)^3).
(Formerly M3125)
+10
7
1, 3, 30, 420, 6930, 126126, 2450448, 49884120, 1051723530, 22787343150, 504636071940, 11377249621920, 260363981732400, 6034149862347600, 141371511060715200, 3343436236585914480, 79726203788589122490, 1914992149823954412750, 46295775130831740013500
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