A260710 Expansion of 1/(1 - x - x^2 - x^4 + x^5 + x^7).
1, 1, 2, 3, 6, 9, 16, 25, 43, 69, 116, 188, 313, 511, 846, 1386, 2288, 3756, 6191, 10174, 16756, 27552, 45357, 74604, 122787, 201996, 332414, 546901, 899946, 1480699, 2436459, 4008858, 6596366, 10853563, 17858788, 29384804, 48350401, 79555943, 130902711
Offset: 0
Examples
There are 25 partially ordered partitions of 7, i.e., a(7) = 25. These are (43=34),(421=412),(124=214),(241),(142),(4111),(1411),(1141),(1114),(331),(313),(133),(1132=1123),(2131=1231),(1312=1321),(2311=3211),(31111),(13111),(11311),(11131),(11113),(2221=four),(22111=ten),(211111=six),(1111111).
Links
- Index entries for related partition-counting sequences
- Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,-1,0,-1).
Programs
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Magma
I:=[1,1,2,3,6,9,16]; [n le 7 select I[n] else Self(n-1)+Self(n-2)+Self(n-4)-Self(n-5)-Self(n-7): n in [1..40]]; // Vincenzo Librandi, Aug 04 2015
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Mathematica
LinearRecurrence[{1, 1, 0, 1, -1, 0, -1}, {1, 1, 2, 3, 6, 9, 16}, 50] (* Vincenzo Librandi, Aug 04 2015 *) -
PARI
Vec(1/(1 - x - x^2 - x^4 + x^5 + x^7) + O(x^50)) \\ Michel Marcus, Aug 06 2015
Formula
G.f: 1/(1 - x - x^2 - x^4 + x^5 + x^7).
a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-5) - a(n-7).
Construct the matrix array T(n,j) = [A^*j]*[S^*(j-1)] where A=(1,1,0,1,-1,0,-1) and S=(0,1,0,...) (A063524). [* is convolution operation] Define S^*0=I with I=(1,0,...). a(n) = Sum_{j=1..n} T(n,j).
Comments