A299337 Expansion of 1 / ((1 - x)^7*(1 + x)^5).
1, 2, 8, 14, 35, 56, 112, 168, 294, 420, 672, 924, 1386, 1848, 2640, 3432, 4719, 6006, 8008, 10010, 13013, 16016, 20384, 24752, 30940, 37128, 45696, 54264, 65892, 77520, 93024, 108528, 128877, 149226, 175560, 201894, 235543, 269192, 311696, 354200, 407330
Offset: 0
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..10000
- Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
- Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).
Programs
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Mathematica
LinearRecurrence[{2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1}, {1, 2, 8, 14, 35, 56, 112, 168, 294, 420, 672, 924}, 41] (* Michael De Vlieger, Dec 19 2024 *) -
PARI
Vec(1 / ((1 - x)^7*(1 + x)^5) + O(x^40))
Formula
a(n) = (2*n^6 + 72*n^5 + 1040*n^4 + 7680*n^3 + 30368*n^2 + 60288*n + 46080) / 46080 for n even.
a(n) = (2*n^6 + 72*n^5 + 1010*n^4 + 6960*n^3 + 24278*n^2 + 39048*n + 20790) / 46080 for n odd.
a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n>11.