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A220212
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Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).
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13
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0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: x*(1+11*x)/(1-x)^5.
a(n) = n*(n+1)*(n+2)*(3*n-2)/6.
Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)
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MATHEMATICA
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A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)
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PROG
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(Magma) A051866:=func<n | n*(6*n-5)>; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];
(Magma) I:=[0, 1, 16, 70, 200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
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CROSSREFS
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Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):
Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.
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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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