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A109128
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Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0<k<n, T(n,0) = T(n,n) = 1.
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23
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1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 11, 7, 1, 1, 9, 19, 19, 9, 1, 1, 11, 29, 39, 29, 11, 1, 1, 13, 41, 69, 69, 41, 13, 1, 1, 15, 55, 111, 139, 111, 55, 15, 1, 1, 17, 71, 167, 251, 251, 167, 71, 17, 1, 1, 19, 89, 239, 419, 503, 419, 239, 89, 19, 1, 1, 21, 109, 329, 659, 923, 923, 659, 329, 109, 21, 1
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 with T(n,0) = T(n,n) = 1.
Sum_{k=0..n} T(n, k) = A000325(n+1) (row sums).
T(n, k) = 2*binomial(n,k) - 1. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Sep 30 2007
T(n, 1) = 2*n - 1 = A005408(n+1) for n>0.
T(n, 2) = n^2 + n - 1 = A028387(n-2) for n>1.
T(n, k) = Sum_{j=0..n-k} C(n-k,j)*C(k,j)*(2 - 0^j) for k <= n. - Paul Barry, Apr 27 2006
T(n, n-k) = T(n, k).
T(2*n-1, n) = A030662(n), for n >= 1.
Sum_{k=0..n-1} T(n, k) = A000295(n+1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = 2*[n=0] - A000035(n+1).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A327767(n), for n >= 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A281362(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A281362(n-1) - (1+(-1)^n)/2 for n >= 1.
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n), where b(n) is the repeating pattern {1,1,0,-2,-3,-1,2,2,-1,-3,-2,0} with b(n) = b(n-12). (End)
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EXAMPLE
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Triangle begins as:
1;
1 1;
1 3 1;
1 5 5 1;
1 7 11 7 1;
1 9 19 19 9 1;
1 11 29 39 29 11 1;
1 13 41 69 69 41 13 1;
1 15 55 111 139 111 55 15 1;
1 17 71 167 251 251 167 71 17 1;
1 19 89 239 419 503 419 239 89 19 1;
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MAPLE
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2*binomial(n, k)-1 ;
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MATHEMATICA
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Table[2*Binomial[n, k] -1, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
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PROG
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(Haskell)
a109128 n k = a109128_tabl !! n !! k
a109128_row n = a109128_tabl !! n
a109128_tabl = iterate (\row -> zipWith (+)
([0] ++ row) (1 : (map (+ 1) $ tail row) ++ [0])) [1]
(Magma) [2*Binomial(n, k) -1: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2020
(Sage) [[2*binomial(n, k) -1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020
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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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