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A094705
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Convolution of Jacobsthal(n) and 3^n.
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5
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0, 1, 4, 15, 50, 161, 504, 1555, 4750, 14421, 43604, 131495, 395850, 1190281, 3576304, 10739835, 32241350, 96767741, 290390604, 871346575, 2614389250, 7843866801, 23532998504, 70601791715, 211810967550, 635444087461
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| For k>2, a(n,k)=k^(n+1)/((k-2)(k+1))-2^(n+1)/(3k-6)-(-1)^n/(3k+3) gives the convolution of Jacobsthal(n) and k^n.
In general x/((1-ax)(1-ax-bx^2)) expands to sum{k=0..floor(n/2), C(n-k,k+1)a^(n-k-1)*(b/a)^k}. - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
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FORMULA
| G.f. : x((1-3x)(1-x-2x^2)); a(n)=3*3^n/4-2*2^n/3-(-1)^n/12; a(n)=4a(n-1)-a(n-2)-6a(n-3).
G.f. : x/((1-2x)(1-2x-3x^2))=x/((1+x)(1-2x)(1-3x)); a(n)=sum{k=0..floor(n/2), binomial(n-k, k+1)2^(n-k-1)(3/2)^k}; - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
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CROSSREFS
| Cf. A001045, A000244, A045883.
Sequence in context: A056327 A026328 A014532 * A196835 A055218 A107307
Adjacent sequences: A094702 A094703 A094704 * A094706 A094707 A094708
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), May 21 2004
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