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A014825
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a(n) = 4*a(n-1) + n with n > 1, a(1)=1.
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21
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1, 6, 27, 112, 453, 1818, 7279, 29124, 116505, 466030, 1864131, 7456536, 29826157, 119304642, 477218583, 1908874348, 7635497409, 30541989654, 122167958635, 488671834560, 1954687338261, 7818749353066
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = (4^(n+1) - 3*n - 4)/9.
G.f.: x/((1-4*x)*(1-x)^2).
a(n) = Sum_{k=0..n} (n-k)*4^k = Sum_{k=0..n} k*4^(n-k). - Paul Barry, Jul 30 2004
a(n) = Sum_{k=0..n} binomial(n+2, k+2)*3^k [Offset 0]. - Paul Barry, Jul 30 2004
a(n) = Sum_{k=0..n} Sum_{j=0..2k} (-1)^(j+1)*J(j)*J(2k-j), J(n) = A001045(n). - Paul Barry, Oct 23 2009
E.g.f.: (4*exp(4*x) - (4+3*x)*exp(x))/9. - G. C. Greubel, Feb 18 2020
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EXAMPLE
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G.f. = x + 6*x^2 + 27*x^3 + 112*x^4 + 453*x^5 + 1818*x^6 + 7279*x^7 + ...
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MATHEMATICA
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RecurrenceTable[{a[1]==1, a[n]==4a[n-1]+n}, a[n], {n, 30}] (* Harvey P. Dale, Oct 12 2011 *)
a[ n_]:= SeriesCoefficient[x/((1-4x)(1-x)^2), {x, 0, n}] (* Michael Somos, Jun 20 2012 *)
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PROG
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(PARI) {a(n) = polcoeff( x / ((1 - x)^2 * (1 - 4*x)) + x * O(x^n), n)} /* Michael Somos, Jun 20 2012 */
(Sage) [(4^(n+1) -3*n -4)/9 for n in (1..30)] # G. C. Greubel, Feb 18 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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