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A224838
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Triangle read by rows, obtained from triangle A011973 by reading that array from right to left along the irregular paths shown in the figure.
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4
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1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 4, 6, 5, 1, 1, 10, 10, 6, 1, 1, 5, 20, 15, 7, 1, 6, 15, 35, 21, 8, 1, 1, 21, 35, 56, 28, 9, 1, 1, 7, 56, 70, 84, 36, 10, 1, 8, 28, 126, 126, 120, 45, 11, 1, 1, 36, 84, 252, 210, 165, 55, 12, 1, 1, 9, 120, 210, 462, 330, 220, 66, 13, 1
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OFFSET
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1,4
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COMMENTS
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The successive rows have lengths 1,2,2; 3,4,4; 5,6,6; 7,8,8; ...
Old definition was: "Triangle of falling diagonals of A011973 (with rows displayed as centered text)."
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LINKS
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FORMULA
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r(n) = binomial(n-floor((4n+15-6k+(-1)^k)/12), n-floor((4n+15-6k+(-1)^k)/12)-floor((2n-1)/3)+k-1), k = 1..floor((2n+2)/3).
R(n) = binomial(n-floor((k+1)/2), n-floor((3k-1)/2)), k = 1..floor((2n+2)/3), gives the terms of each row in reverse order.
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EXAMPLE
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First 11 rows of the triangle:
1;
1, 1;
2, 1;
1, 3, 1;
1, 3, 4, 1;
4, 6, 5, 1;
1, 10, 10, 6, 1;
1, 5, 20, 15, 7, 1;
6, 15, 35, 21, 8, 1;
1, 21, 35, 56, 28, 9, 1;
1, 7, 56, 70, 84, 36, 10, 1;
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MATHEMATICA
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Table[Reverse[Table[Binomial[n - Floor[(k + 1)/2], n - Floor[(3 k - 1)/2]], {k, Floor[(2 n + 2)/3]}]], {n, 13}] (* T. D. Noe, Jul 25 2013 *)
Column[Table[Binomial[n - Floor[(4 n + 15 - 6 k + (-1)^k)/12], n - Floor[(4 n + 15 - 6 k + (-1)^k)/12] - Floor[(2 n - 1)/3] + k - 1], {n, 1, 25}, {k, 1, Floor[(2 n + 2)/3]}]] (* John Molokach, Jul 25 2013 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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