Displaying 1-10 of 31 results found.
Square array A(n,k) for n,k>=0, where A(n,k) is the number of paths from (0,0) to (n,k) in the directed graph with vertices (i,j) and edges (i,j)-to-(i+1,j), (i,j)-to-(i,j+1), and (i,i+h)-to-(i+1,i+h+1) for every i,j,h>=0.
+0
1
1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 5, 12, 7, 1, 1, 6, 17, 24, 9, 1, 1, 7, 23, 53, 40, 11, 1, 1, 8, 30, 76, 117, 60, 13, 1, 1, 9, 38, 106, 246, 217, 84, 15, 1, 1, 10, 47, 144, 352, 580, 361, 112, 17, 1, 1, 11, 57, 191, 496, 1178, 1158, 557, 144, 19, 1, 1, 12, 68, 248, 687, 1674, 2916, 2076, 813, 180, 21, 1, 1, 13, 80, 316, 935, 2361, 5768, 6150, 3446, 1137, 220, 23, 1, 1, 14, 93
FORMULA
For n>=2*k, A(n,k) = coefficient of x^k in F(x)*C(x)^(n-2*k). For n<=2*k, A(n,k) = coefficient of x^(n-k) in F(x)*S(x)^(2*k-n). Here C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318, and F(x)=S(x)/(1-x*C(x)*S(x)) is o.g.f. for A026781.
CROSSREFS
Row-reversed or transposed version of A026780.
1, 19, 220, 2021, 16246, 119914, 835080, 5579147, 36150820, 228879616, 1423367610, 8727874180, 52919787978, 317969615146, 1896431659490, 11241964305405, 66305996581006, 389432443869294, 2279159225280808
1, 15, 144, 1137, 8070, 53678, 342224, 2119359, 12856752, 76826324, 453947206, 2659486160, 15479103062, 89636939038, 517014933254, 2972784038825, 17051188983758, 97611767361074, 557935896092592, 3185250335024206
1, 5, 23, 106, 496, 2361, 11425, 56128, 279530, 1409242, 7182570, 36965292, 191896092, 1003889999, 5288020783, 28026601758, 149362212188, 799947847992, 4303496405324, 23245282077346, 126020035603358, 685477891161324
1, 7, 38, 191, 935, 4547, 22166, 108790, 538668, 2692996, 13594954, 69281309, 356222591, 1846781939, 9647229202, 50744420578, 268591503348, 1429730471552, 7649564591070, 41117462702680, 221938325574492, 1202501072716220
1, 9, 57, 316, 1647, 8330, 41558, 206418, 1026392, 5126109, 25763881, 130446758, 665644541, 3423360416, 17740450004, 92599616864, 486607242642, 2573051443194, 13683364565774, 73146530405912, 392867752273562, 2119154966713452
1, 11, 80, 489, 2732, 14512, 74888, 380329, 1916709, 9637687, 48528434, 245289395, 1246499970, 6374296278, 32816505566, 170106486358, 887697088316, 4662338571684, 24636365887246, 130918389450108, 699335986465354
1, 6, 30, 144, 687, 3296, 15972, 78294, 388320, 1947910, 9875566, 50560246, 261177401, 1360112590, 7134802722, 37673830960, 200106632766, 1068539351340, 5733226876876, 30894846668416, 167137498306038, 907416216735816
1, 8, 47, 248, 1251, 6194, 30496, 150348, 745086, 3719388, 18721063, 95045190, 486669349, 2512426480, 13070589618, 68484870582, 361191120212, 1916337714194, 10222616034264, 54800827268454, 295084855980404, 1595368824989782
1, 10, 68, 396, 2136, 11062, 56070, 281306, 1406721, 7042818, 35401568, 178975192, 910933936, 4669860386, 24114746282, 125416122430, 656713729000, 3460748531510, 18345703137458, 97782896293158, 523786141723670
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