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A053581
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First differences of the Poly-Bernoulli numbers B_n^(k) with k=-2 (A027649).
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5
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1, 3, 10, 32, 100, 308, 940, 2852, 8620, 25988, 78220, 235172, 706540, 2121668, 6369100, 19115492, 57362860, 172121348, 516429580, 1549419812, 4648521580, 13946089028, 41839315660, 125520044132
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OFFSET
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0,2
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COMMENTS
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Also the second differences of A001047.
Equals sum of "terms added" to current row of the triangle version of A038573 to get the next row. a(3) = 32 sum of (3, 7, 7, 15) = terms appended to row 2 of the triangle in A038573. - Gary W. Adamson, Jun 04 2009
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LINKS
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FORMULA
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a(n) = 5*a(n-1) - 6*a(n-2) + C(2,2-n), n>1, with a(0)=1, a(1)=3, where C(2, 2-n)=1 for n=2 and =0 for n>2.
Binomial transform of A000975(n+1).
G.f.: (1-x)^2/((1-2*x)*(1-3*x)).
a(n) = 4*3^n/3 + 0^n/6 - 2^n/2. (End)
a(n) = Sum_{k=0..n+1} binomial(n+1, k) * Sum_{j=0..floor(k/2)} A001045(k-2*j). - Paul Barry, Apr 17 2005
E.g.f.: (1 - 3*exp(2*x) + 8*exp(3*x))/6. - G. C. Greubel, May 16 2019
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MATHEMATICA
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CoefficientList[Series[(1-x)^2/((1-2x)(1-3x)), {x, 0, 30}], x] (* Harvey P. Dale, Apr 22 2011 *)
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PROG
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(PARI) vector(30, n, n--; 4*3^(n-1) +(0^n -3*2^n)/6) \\ G. C. Greubel, May 16 2019
(Sage) [4*3^(n-1) +(0^n -3*2^n)/6 for n in (0..30)] # G. C. Greubel, May 16 2019
(GAP) List([0..30], n-> 4*3^(n-1) +(0^n -3*2^n)/6) # G. C. Greubel, May 16 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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