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A134346
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Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).
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5
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1, 3, 3, 7, 14, 7, 15, 45, 45, 15, 31, 124, 186, 124, 31, 63, 315, 630, 630, 315, 63, 127, 762, 1905, 2540, 1905, 762, 127, 255, 1785, 5355, 8925, 8925, 5355, 1785, 255, 511, 4088, 14308, 28616, 35770, 28616, 14308, 4088, 511
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OFFSET
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0,2
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COMMENTS
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Inverse binomial transform: A134347.
Also the triangle t with offset 1 and elements t(n, m) = T(n-1, m-1) read by rows, giving in row n >= 1 the sums of the entries of A356028 of like m.
Also triangle t with offset 1 read by rows, giving in row n >= 1 the sum of the numbers from 1, 2, ..., 2^n - 1 with binary weight m, for m = 1, 2, ..., n. [Observation by Kevin Ryde.] (End)
T(n,k) is the sum of the entries in the (k+2)-th column of the Christmas tree pattern (A367562) of order n+1. - Paolo Xausa, Dec 20 2023
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LINKS
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FORMULA
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T(n, k) = 0 for n < k, T(n, 0) = 2^(n+1) - 1, and
T(n, k) = T(n-1, k) + T(n-1, k-1) + binomial(n, k)*2^n, or
T(n, k) = 2*(T(n-1, k) + T(n-1, k-1)) + binomial(n-1, k-1).
(Proof for T(n-1, m-1) = t(n, m), offset 1, by separating in the list of the binary code of the numbers 1, 2, ..., 2^n-1 of length n and weight m the sublists with first entry 1 and 0. The total number of elements of the list for n and m is binomial(n, m).) (End)
T(n, k) = [x^k] ((1/2 - x)^(k - n - 1) - (1 - x)^(k - n - 1)). - Peter Luschny, Aug 22 2022
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EXAMPLE
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First few rows of the triangle:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 3 3
2: 7 14 7
3: 15 45 45 15
4: 31 124 186 124 31
5: 63 315 630 630 315 63
6: 127 762 1905 2540 1905 762 127
7: 255 1785 5355 8925 8925 5355 1785 255
8: 511 4088 14308 28616 35770 28616 14308 4088 511
9: 1023 9207 36828 85932 128898 128898 85932 36828 9207 1023
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T(3, 1) = 12 + 10 + 9 + 6 + 5 + 3 = 45. (From A356028 row n = 4, m = 2.)
Recurrences: T(4, 1) = 45 + 15 + 4*16 = 2*(45 + 15) +4 = 124. - Wolfdieter Lang, Jul 27 2022
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MAPLE
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(2^(n+1)-1)*binomial(n, k) ;
end proc:
ser := series((1/2 - x)^(k - n - 1) - (1 - x)^(k - n - 1), x, 10):
seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9); # Peter Luschny, Aug 22 2022
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MATHEMATICA
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A134346[n_, k_]:=(2^(n+1)-1)Binomial[n, k];
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PROG
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(PARI) T(n, k) = my(b=binomial(n, k)); b<<(n+1) - b; \\ Kevin Ryde, Aug 15 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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