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A366120
Number of ways to choose a multiset of values whose sum is n from the first floor(1+log_2(n)) rows of Pascal's triangle.
0
1, 3, 1, 15, 11, 5, 1, 141, 127, 100, 70, 44, 22, 7, 1, 2453, 2393, 2249, 2035, 1784, 1508, 1222, 948, 694, 472, 302, 182, 95, 37, 9, 1, 84870, 84278, 82769, 80217, 76934, 73302, 69363, 64917, 59951, 54787, 49746, 44826, 39850, 34830, 30018, 25614, 21575, 17777
OFFSET
1,2
FORMULA
a(n) = [x^n] Product_{k=0..floor(log_2(n))} Product_{j=0..k} (1 + x^binomial(k, j)). - Andrew Howroyd, Oct 28 2023
EXAMPLE
For n = 5 we use the first 3 rows of Pascal's triangle:
1;
1, 1;
1, 2, 1;
There are C(1,1)*C(5,3) = 10 ways to make the sum 2 + 1 + 1 + 1
and C(5,5) = 1 ways to make the sum 1 + 1 + 1 + 1 + 1,
so a(5) = 10 + 1 = 11.
PROG
(PARI) a(n) = polcoef(prod(k=0, logint(n, 2), prod(j=0, k, 1 + x^binomial(k, j) + O(x*x^n))), n) \\ Andrew Howroyd, Oct 28 2023
CROSSREFS
Cf. A007318 (Pascal's triangle).
Sequence in context: A144006 A014621 A193966 * A113378 A365162 A178657
KEYWORD
nonn
AUTHOR
Guderian Raborg, Sep 30 2023
EXTENSIONS
Terms a(16) and beyond from Andrew Howroyd, Oct 28 2023
STATUS
approved