Revision History for A134319
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#25 by Michael De Vlieger at Fri Jun 23 10:31:44 EDT 2023
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#24 by Peter Luschny at Fri Jun 23 10:05:16 EDT 2023
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#23 by Peter Luschny at Fri Jun 23 10:01:13 EDT 2023
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| MAPLE
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Trow := n -> seq((-1)^k * coeff(p(n, 0), x, n - k), k = 0..n):): # _Peter Luschny_, Jun 23 2023
seq(Trow(n), n = 0..10); # Peter Luschny, Jun 23 2023
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| CROSSREFS
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Cf. A083313, A083323 (row sums), A255047 (main diagonal).
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#22 by Peter Luschny at Fri Jun 23 09:53:23 EDT 2023
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| NAME
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A007318 * aTriangle triangleread by rows: for . T(n > 0, k) = binomial(n zeros followed by , k)*(2^nk - 1. + 0^k).
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| FORMULA
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Previous definition: A007318 * a triangle by rows: for n > 0, n zeros followed by 2^n - 1.
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| MAPLE
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T := (n, k) -> binomial(n, k)*(2^k - 1 + 0^k):
for n from 0 to 7 do seq(T(n, k), k=0..n) od;
# Or as a recursion:
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| EXTENSIONS
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New name using a formula of Yuchun Ji by Peter Luschny, Jun 23 2023
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#21 by Peter Luschny at Fri Jun 23 09:37:49 EDT 2023
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| COMMENTS
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Row sums = A083323: (1, 2, 6, 20, 66, 212, 666, ...).
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| MAPLE
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# Alternative:
p := proc(n, m) option remember; if n = 0 then max(1, m) else
(m + x)*p(n - 1, m) - (m + 1)*p(n - 1, m + 1) fi end:
Trow := n -> seq((-1)^k * coeff(p(n, 0), x, n - k), k = 0..n):
seq(Trow(n), n = 0..10); # Peter Luschny, Jun 23 2023
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| CROSSREFS
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Cf. A083313., A083323 (row sums).
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| STATUS
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approved
editing
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#20 by N. J. A. Sloane at Sat Feb 16 06:57:13 EST 2019
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#19 by Jon E. Schoenfield at Tue Feb 12 22:25:18 EST 2019
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Discussion
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| Wed Feb 13
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| Michel Marcus: Maybe I made a mistake, but for me the new formula does not work
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| 03:47
| Michel Marcus: No , my mistake M(n) has offset 0 with M(k) = if (k<1, 1, 2^k-1)
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| 19:39
| Yuchun Ji: @Michel Marcus, Alois P. Heinz said (1, 1, 3, 7, 15, ...) just is the Mersenne-like A255047, called M(k) here. For example T(3,k) = (1, 3, 9, 7) = (1*1, 3*1, 3*3, 1*7), where C(3,k) = (1, 3, 3, 1) and M(k) = (1, 1, 3, 7).
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#18 by Jon E. Schoenfield at Tue Feb 12 22:25:15 EST 2019
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| NAME
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A007318 * a triangle by rows: for n> > 0, n zeros followed by 2^n - 1.
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| COMMENTS
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Row sums = A083323: (1, 2, 6, 20, 66, 212, 666,...)., ...).
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| FORMULA
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Binomial transform of a diagonalized infinite lower triangular matrix with (1, 1, 3, 7, 15,...) , ...) in the main diagonal and the rest zeros.
T(n,k) = |[1/(2^x)^k] 1+( + (1-1/2^x)^n-( - (1-2/2^x)^n|. - Alois P. Heinz, Dec 10 2008
T(n,k) = Cbinomial(n,k)*M(k). ) where C is binomial, M is Mersenne-like A255047. - Yuchun Ji, Feb 13 2019.
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| EXAMPLE
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First few rows of the triangle are:
1;
1, 1;
1, 2, 3;
1, 3, 9, 7;
1, 4, 18, 28, 15;
1, 5, 30, 70, 75, 31;
1, 6, 45, 140, 225, 186, 63;
1, 7, 63, 245, 525, 651, 441, 127;
...
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| STATUS
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proposed
editing
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#17 by Yuchun Ji at Tue Feb 12 21:36:08 EST 2019
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#16 by Yuchun Ji at Tue Feb 12 21:34:43 EST 2019
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| FORMULA
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T(n,k) = C(n,k)*M(k) for k > 0, T(n,0) = 1, ). where C is binomial, M is Mersenne-like A000225A255047. - Yuchun Ji, Feb 13 2019.
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