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A367076
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Irregular triangle read by rows: T(n,k) (0 <= n, 0 <= k < 2^n). T(n,k) = -Sum_{i=0..k} A365968(n,i).
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0
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0, 1, 0, 3, 4, 3, 0, 6, 10, 12, 12, 12, 10, 6, 0, 10, 18, 24, 28, 32, 34, 34, 32, 34, 34, 32, 28, 24, 18, 10, 0, 15, 28, 39, 48, 57, 64, 69, 72, 79, 84, 87, 88, 89, 88, 85, 80, 85, 88, 89, 88, 87, 84, 79, 72, 69, 64, 57, 48, 39, 28, 15, 0, 21, 40, 57, 72, 87
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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LINKS
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FORMULA
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T(n,k) = Sum_{i=0..n} abs(k + 1 - (2^i) * round((k+1)/2^i)) * i.
G.f. for n-th row: 1/(1-x) * Sum_{i=1..n} (i/(1+x^2^(i-1)) * Product_{j=0..i-2} 1 + x^2^j).
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EXAMPLE
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Triangle begins:
k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
n=0: 0;
n=1: 1, 0;
n=2: 3, 4, 3, 0;
n=3: 6, 10, 12, 12, 12, 10, 6, 0;
n=4; 10, 18, 24, 28, 32, 34, 34, 32, 34, 34, 32, 28, 24, 18, 10, 0;
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MATHEMATICA
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nmax=10; row[n_]:=Join[CoefficientList[Series[1/(1-x)*Sum[ i/(1+x^2^(i-1))*Product[1+x^2^j, {j, 0, i-2}], {i, n}], {x, 0, 2^n-1}], x], {0}]; Array[row, 6, 0] (* Stefano Spezia, Dec 23 2023 *)
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PROG
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(Python)
def row_gen(n):
x = 0
for k in range(2**n):
b = bin(k)[2:].zfill(n)
x += sum((-1)**(int(b[n-i])+1)*i for i in range(1, n+1))
yield(-x)
def A367076_row_n(n): return(list(row_gen(n)))
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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