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A355664
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Square array A(n, k), n, k >= 0, read by antidiagonals; for any number n with runs in binary expansion (r_w, ..., r_0), let p(n) be the polynomial of a single indeterminate x where the coefficient of x^e is r_e for e = 0..w and otherwise 0, and let q be the inverse of p; A(n, k) = q(p(n) * p(k)).
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1
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0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 9, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 35, 15, 35, 5, 0, 0, 6, 38, 48, 48, 38, 6, 0, 0, 7, 49, 51, 271, 51, 49, 7, 0, 0, 8, 56, 60, 284, 284, 60, 56, 8, 0, 0, 9, 135, 63, 387, 313, 387, 63, 135, 9, 0, 0, 10, 142, 192, 448, 398, 398, 448, 192, 142, 10, 0
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OFFSET
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0,8
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COMMENTS
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In other words, A(n, k) encodes the product of the polynomials encoded by n and k.
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LINKS
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FORMULA
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A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, 1) = n.
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EXAMPLE
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Array A(n, k) begins:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11
---+---------------------------------------------------------------------
0| 0 0 0 0 0 0 0 0 0 0 0 0
1| 0 1 2 3 4 5 6 7 8 9 10 11
2| 0 2 9 12 35 38 49 56 135 142 153 156
3| 0 3 12 15 48 51 60 63 192 195 204 207
4| 0 4 35 48 271 284 387 448 2111 2172 2275 2288
5| 0 5 38 51 284 313 398 455 2168 2289 2502 2531
6| 0 6 49 60 387 398 481 504 3079 3102 3185 3196
7| 0 7 56 63 448 455 504 511 3584 3591 3640 3647
8| 0 8 135 192 2111 2168 3079 3584 33279 33784 34695 34752
9| 0 9 142 195 2172 2289 3102 3591 33784 34785 36622 36739
10| 0 10 153 204 2275 2502 3185 3640 34695 36622 39993 40476
11| 0 11 156 207 2288 2531 3196 3647 34752 36739 40476 40719
12| 0 12 195 240 3087 3132 3843 4032 49215 49404 50115 50160
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PROG
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(PARI) toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
A(n, k) = { fromruns(Vec(Pol(toruns(n)) * Pol(toruns(k)))) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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