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A130154
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Triangle read by rows: T(n, k) = 1 + 2*(n-k)*(k-1) (1 <= k <= n).
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5
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1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 9, 7, 1, 1, 9, 13, 13, 9, 1, 1, 11, 17, 19, 17, 11, 1, 1, 13, 21, 25, 25, 21, 13, 1, 1, 15, 25, 31, 33, 31, 25, 15, 1, 1, 17, 29, 37, 41, 41, 37, 29, 17, 1, 1, 19, 33, 43, 49, 51, 49, 43, 33, 19, 1, 1, 21, 37, 49, 57, 61, 61, 57, 49, 37, 21, 1
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OFFSET
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1,5
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COMMENTS
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Column k, except for the initial k-1 0's, is an arithmetic progression with first term 1 and common difference 2(k-1). Row sums yield A116731. First column of the inverse matrix is A129779.
T(n, k) gives the number of distinct sums of 2(k-1) elements in {1,1,2,2,...,n-1,n-1}. For example, T(6, 2) = the number of distinct sums of 2 elements in {1,1,2,2,3,3,4,4,5,5}, and because each sum from the smallest 1 + 1 = 2 to the largest 5 + 5 = 10 appears, T(6, 2) = 10 - 1 = 9. [In general:
2*Sum_{j=1..(k-1)} (n-j) - (2*(Sum_{j=1..k-1} j) - 1) = 2*(n*(k-1) - 4*(k-1)*k/2 + 1 = 2*(k-1)*(n-k) + 1 = T(n, k). - Wolfdieter Lang, Dec 20 2017]
T(n, k) is the number of lattice points with abscissa x = 2*(k-1) and even ordinate in the closed region bounded by the parabola y = x*(2*(n-1) - x) and the x axis. [That is, (1/2)*y(2*(k-1)) + 1 = T(n, k). - Wolfdieter Lang, Dec 20 2017]
Pascal's triangle (A007318, but with apex in the middle) is formed using the rule South = West + East; the rascal triangle A077028 uses the rule South = (West*East + 1)/North; the present triangle uses a similar rule: South = (West*East + 2)/North. See the formula section for this recurrence. (End)
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LINKS
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FORMULA
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T(n, k) = 1 + 2*(n-k)*(k-1) (1 <= k <= n).
G.f.: G(t,z) = t*z*(3*t*z^2 - z - t*z + 1)/((1-t*z)*(1-z))^2.
T(n, 1) = 1 and T(n, n) = 1 for n >= 1; T(n, k) = (T(n-1, k-1)*T(n-1, k) + 2)/T(n-2, k-1), for n > 2 and 1 < k < n. See a comment above. - Rogério Serôdio, Dec 19 2017
G.f. column k (with leading zeros): (x^k/(1-x)^2)*(1 + (2*k-3)*x), k >= 1. See the g.f. of the triangle G(t,z) above: (d/dt)^k G(t,x)/k!|_{t=0}. - Wolfdieter Lang, Dec 20 2017
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EXAMPLE
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The triangle T(n, k) starts:
n\k 1 2 3 4 5 6 7 8 9 10 ...
1: 1
2: 1 1
3: 1 3 1
4: 1 5 5 1
5: 1 7 9 7 1
6: 1 9 13 13 9 1
7: 1 11 17 19 17 11 1
8: 1 13 21 25 25 21 13 1
9: 1 15 25 31 33 31 25 15 1
10: 1 17 29 37 41 41 37 29 17 1
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MAPLE
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T:=proc(n, k) if k<=n then 2*(n-k)*(k-1)+1 else 0 fi end: for n from 1 to 14 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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MATHEMATICA
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Flatten[Table[1+2(n-k)(k-1), {n, 0, 20}, {k, n}]] (* Harvey P. Dale, Jul 13 2013 *)
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PROG
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(PARI) T(n, k) = 1 + 2*(n-k)*(k-1) \\ Iain Fox, Dec 19 2017
(PARI) first(n) = my(res = vector(binomial(n+1, 2)), i = 1); for(r=1, n, for(k=1, r, res[i] = 1 + 2*(r-k)*(k-1); i++)); res \\ Iain Fox, Dec 19 2017
(Magma) [1 + 2*(n-k)*(k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 25 2019
(Sage) [[1 + 2*(n-k)*(k-1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 25 2019
(GAP) Flat(List([1..12], n-> List([1..n], k-> 1 + 2*(n-k)*(k-1) ))); # G. C. Greubel, Nov 25 2019
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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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