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A152920
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Triangle read by rows: triangle A062111 reversed.
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12
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0, 1, 1, 2, 3, 4, 3, 5, 8, 12, 4, 7, 12, 20, 32, 5, 9, 16, 28, 48, 80, 6, 11, 20, 36, 64, 112, 192, 7, 13, 24, 44, 80, 144, 256, 448, 8, 15, 28, 52, 96, 176, 320, 576, 1024, 9, 17, 32, 60, 112, 208, 384, 704, 1280, 2304, 10, 19, 36, 68, 128, 240, 448, 832, 1536, 2816, 5120
(list;
table;
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) = (2*n-k) * 2^(k-1) for 0 <= k <= n.
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = t*(1+x-3*x*t) / ((1-t)^2 * (1-2*x*t)^2).
Sum_{k=0..n} (-1)^k * binomial(n,k) * T(n,k) = 0 for n >= 0.
Sum_{k=0..n} binomial(n,k) * T(n,k) = 2*n * 3^(n-1) for n >= 0.
Define the array B(n,p) = (Sum_{k=0..n} binomial(p+k,p) * T(n,k))/(n+p+1) for n >= 0 and p >= 0. Then see the comment of Robert Coquereaux (2014) at A193844. Conjecture: B(n+1,p) = A(n,p). (End)
T(n, k) = T(n, k-1) + T(n-1, k-1) for k>=1, T(n,0) = n. - Alois P. Heinz, Sep 12 2022
T(m*n, n) = (2*m-1)*A001787(n), for m >= 1. (End)
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EXAMPLE
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Triangle starts:
0;
1, 1;
2, 3, 4;
3, 5, 8, 12;
4, 7, 12, 20, 32;
...
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MAPLE
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A062111 := proc(n, k) (k+n)*2^(k-n-1) ; end: A152920 := proc(n, k) A062111(n-k, n) ; end: for n from 0 to 15 do for k from 0 to n do printf("%d, ", A152920(n, k)) ; od: od: # R. J. Mathar, Jan 22 2009
# second Maple program:
T:= proc(n, k) option remember;
`if`(k=0, n, T(n, k-1)+T(n-1, k-1))
end:
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MATHEMATICA
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t[0, k_]:= k; t[n_, k_]:= t[n, k]= t[n-1, k] + t[n-1, k+1];
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PROG
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(Magma) [2^k*(n-k/2): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2022
(SageMath) flatten([[2^(k-1)*(2*n-k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 27 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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