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A228888
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a(n) = binomial(3*n + 2, 3).
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6
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10, 56, 165, 364, 680, 1140, 1771, 2600, 3654, 4960, 6545, 8436, 10660, 13244, 16215, 19600, 23426, 27720, 32509, 37820, 43680, 50116, 57155, 64824, 73150, 82160, 91881, 102340, 113564, 125580, 138415, 152096, 166650, 182104, 198485, 215820, 234136, 253460
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OFFSET
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1,1
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LINKS
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FORMULA
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a(n) = binomial(3*n + 2, 3) = 1/6*(3*n)*(3*n + 1)*(3*n + 2).
G.f.: (10*x + 16*x^2 + x^3)/(1 - x)^4 = 10*x + 56*x^2 + 165*x^3 + ....
Sum {n >= 1} 1/a(n) = 9/2 - 3/2*log(3) - 1/2*sqrt(3)*Pi.
Sum {n >= 1} (-1)^n/a(n) = 9/2 - 4*log(2) - 1/3*sqrt(3)*Pi.
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EXAMPLE
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Including 0, row sums of the triangle:
| 0| .................................................................. 0
| 1| 2 3 4 ..................................................... 10
| 5| 6 7 8 9 10 11 ......................................... 56
|12| 13 14 15 16 17 18 19 20 21 ............................ 165
|22| 23 24 25 26 27 28 29 30 31 32 33 34 ................ 364
|35| 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 .... 680
...
in the first column of which we have the pentagonal numbers (A000326).
(End)
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MAPLE
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seq(binomial(3*n+2, 3), n = 1..38);
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MATHEMATICA
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PROG
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CROSSREFS
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Similar sequences are listed in A316224.
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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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