A175939
Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|k>0,0
Original entry on oeis.org
1, 2, 10, 62, 448, 3495, 28640, 242946, 2114829, 18783658, 169546150, 1550728135, 14340859992, 133867779775, 1259689173181, 11936488052113, 113799596287017, 1090803942244627, 10505978544362607, 101623141479156708, 986801698075230291
Offset: 0
- J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From N. J. A. Sloane, Dec 27 2012
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Flatten[{1,RecurrenceTable[{(n+2)*a[n]+(3*n+7)*a[n+1]-(5*n+24)*a[n+2]-(19*n+90)*a[n+3]+(n+3)*a[n+4]+(21*n+116)*a[n+5]-2*(n+7)*a[n+6]==0, a[1]==2, a[2]==10, a[3]==62, a[4]==448, a[5]==3495, a[6]==28640},a,{n,20}]}] (* Vaclav Kotesovec, Sep 07 2012 *)
A175935
Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|0
Original entry on oeis.org
1, 2, 10, 55, 351, 2401, 17248, 128221, 978082, 7612155, 60204488, 482481220, 3909460725, 31974923487, 263623879118, 2188682538746, 18282238300443, 153537981720402, 1295640515428649, 10980400434511117, 93418283866708579
Offset: 0
A175936
Number of lattice walks from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|0
Original entry on oeis.org
1, 2, 10, 62, 433, 3262, 25784, 210892, 1769793, 15152252, 131826824, 1162114368, 10357863128, 93183955872, 845064072102, 7717150002692, 70903816529979, 654967192303546, 6079243786794502, 56668633625876866, 530291242720187193
Offset: 0
A175937
Number of lattice paths from (0,0) to (n,n) using steps S={(k,0),(0,k),(r,r)|0
Original entry on oeis.org
1, 2, 10, 62, 448, 3464, 28111, 236022, 2033145, 17867442, 159558635, 1443747386, 13207922431, 121962046864, 1135246916024, 10640772522150, 100346005711723, 951400275042466, 9063703952844960, 86718277215053218, 832901296331740527
Offset: 0
A307733
a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-1} a(k) * a(n-k-1).
Original entry on oeis.org
1, 1, 4, 14, 54, 220, 934, 4090, 18344, 83850, 389214, 1829736, 8693962, 41685714, 201442188, 980091814, 4797070022, 23603701828, 116688837886, 579312087802, 2887020896016, 14437318756818, 72424982972862, 364366674463824, 1837954750285458
Offset: 0
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a[0] = a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
nmax = 24; CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 2 x^3 + x^4])/(2 x), {x, 0, nmax}], x]
A346506
G.f. A(x) satisfies: A(x) = (1 + x * A(x)^2) / (1 - x + x^2).
Original entry on oeis.org
1, 2, 5, 17, 66, 274, 1190, 5341, 24577, 115326, 549747, 2654739, 12959468, 63848307, 317064921, 1585380283, 7975134892, 40332823042, 204947059412, 1045859173864, 5357606584326, 27540884494209, 142023060613755, 734506610474205, 3808771672620618, 19798640525731461, 103149287155802941
Offset: 0
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nmax = 26; A[] = 0; Do[A[x] = (1 + x A[x]^2)/(1 - x + x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 2; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 1], {k, 2, n - 1}]; Table[a[n], {n, 0, 26}]
Showing 1-6 of 6 results.