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A185787
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Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.
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16
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1, 7, 25, 62, 125, 221, 357, 540, 777, 1075, 1441, 1882, 2405, 3017, 3725, 4536, 5457, 6495, 7657, 8950, 10381, 11957, 13685, 15572, 17625, 19851, 22257, 24850, 27637, 30625, 33821, 37232, 40865, 44727, 48825, 53166, 57757, 62605, 67717, 73100, 78761, 84707, 90945, 97482, 104325, 111481, 118957, 126760, 134897, 143375
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OFFSET
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1,2
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COMMENTS
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This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
1....2.....4.....7...11...16...22...29...
3....5.....8....12...17...23...30...38...
6....9....13....18...24...31...39...48...
10...14...19....25...32...40...49...59...
15...20...26....33...41...50...60...71...
21...27...34....42...51...61...72...84...
28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787. Deleting the main diagonal and then summing give A185787. Analogous treatments to the left of the main diagonal give A100182 and A101165. Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
Let S(n,k) denote the n-th partial sum of column k. Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509
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LINKS
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FORMULA
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a(n)=n*(7*n^2-6*n+5)/6.
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MATHEMATICA
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f[n_, k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n, k], {n, 1, k}];
Factor[s[k]]
Table[s[k], {k, 1, 70}] (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 04 2012 *)
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PROG
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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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EXTENSIONS
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STATUS
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approved
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