A355664 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)).

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

Offset

0,8

Comments

In other words, A(n, k) encodes the product of the polynomials encoded by n and k.

Links

Table of n, a(n) for n=0..77.

Index entries for sequences related to binary expansion of n

Formula

A(n, k) = A(k, n).

A(n, 0) = 0.

A(n, 1) = n.

A(n, 3) = A001196(n).

A(n, 7) = A097254(n+1).

A(n, n) = A355654(n).

Example

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

Prog

(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)))) }

Crossrefs

Cf. A001196, A097254, A101211, A355663.

Sequence in context: A351786 A329331 A370049 * A254040 A376725 A062275

Adjacent sequences: A355661 A355662 A355663 * A355665 A355666 A355667

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Jul 13 2022

Status

approved