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A210381
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Triangle by rows, derived from the beheaded Pascal's triangle, A074909.
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1
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1, 0, 2, 0, 1, 3, 0, 1, 3, 4, 0, 1, 4, 6, 5, 0, 1, 5, 10, 10, 6, 0, 1, 6, 15, 20, 15, 7, 0, 1, 7, 21, 35, 35, 21, 8, 0, 1, 8, 28, 56, 70, 56, 28, 9, 0, 1, 9, 36, 84, 126, 126, 84, 36, 10, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11
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OFFSET
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0,3
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COMMENTS
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Row sums of the triangle = 2^n.
Let the triangle = an infinite lower triangular matrix, M. Then M * The Bernoulli numbers, A027641/A027642 as a vector V = [1, -1, 0, 0, 0,...]. M * the Bernoulli sequence variant starting [1, 1/2, 1/6,...] = [1, 1, 1,...]. M * 2^n: [1, 2, 4, 8,...] = A027649. M * 3^n = A255463; while M * [1, 2, 3,...] = A047859, and M * A027649 = A027650.
Row sums of powers of the triangle generate the Poly-Bernoulli number sequences shown in the array of A099594. - Gary W. Adamson, Mar 21 2012
Triangle T(n,k) given by (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012
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REFERENCES
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Konrad Knopp, Elements of the Theory of Functions, Dover, 1952,pp 117-118.
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LINKS
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FORMULA
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Partial differences of the beheaded Pascal's triangle A074909 starting from the top, by columns.
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,2) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012
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EXAMPLE
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{1},
{0, 2},
{0, 1, 3},
{0, 1, 3, 4},
{0, 1, 4, 6, 5},
{0, 1, 5, 10, 10, 6},
{0, 1, 6, 15, 20, 15, 7},
{0, 1, 7, 21, 35, 35, 21, 8},
{0, 1, 8, 28, 56, 70, 56, 28, 9},
{0, 1, 9, 36, 84, 126, 126, 84, 36, 10},
{0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11}
...
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MATHEMATICA
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t2[n_, m_] = If[m - 1 <= n, Binomial[n, m - 1], 0];
O2 = Table[Table[If[n == m, t2[n, m] + 1, t2[n, m]], {m, 0, n}], {n, 0, 10}];
Flatten[O2]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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