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 A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals. 12
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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: row 1:  A000124 row 2:  A022856 row 3:  A016028 row 4:  A145018 row 5:  A077169 col 1:  A000217 col 2:  A000096 col 3:  A034856 col 4:  A055998 col 5:  A046691 col 6:  A052905 col 7:  A055999 diag. (1,5,...) ...... A001844 diag. (2,8,...) ...... A001105 diag. (4,12,...)...... A046092 diag. (7,17,...)...... A056220 diag. (11,23,...) .... A132209 diag. (16,30,...) .... A054000 diag. (22,38,...) .... A090288 diag. (3,9,...) ...... A058331 diag. (6,14,...) ..... A051890 diag. (10,20,...) .... A005893 diag. (15,27,...) .... A097080 diag. (21,35,...) .... A093328 diag. (28,44,...) .... A147882 antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817. 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 Weight array of T: A144112 Accumulation array of T: A185506 Second rectangular sum array of T: A185507 Third rectangular sum array of T: A185508 Fourth rectangular sum array of T: A185509 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA a(n)=n*(7*n^2-6*n+5)/6. G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012 MATHEMATICA 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 *) PROG (MAGMA) [n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012 CROSSREFS Cf. A000027, A185788, A100182, A101165, A079824. Sequence in context: A118395 A118396 A193375 * A299273 A001845 A127765 Adjacent sequences:  A185784 A185785 A185786 * A185788 A185789 A185790 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 03 2011 STATUS approved

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Last modified December 12 03:27 EST 2018. Contains 318052 sequences. (Running on oeis4.)