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A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.

%I A185787 #30 Feb 26 2023 08:18:54
%S A185787 1,7,25,62,125,221,357,540,777,1075,1441,1882,2405,3017,3725,4536,
%T A185787 5457,6495,7657,8950,10381,11957,13685,15572,17625,19851,22257,24850,
%U A185787 27637,30625,33821,37232,40865,44727,48825,53166,57757,62605,67717,73100,78761,84707,90945,97482,104325,111481,118957,126760,134897,143375
%N A185787 Sum of first k numbers in column k of the natural number array A000027; by antidiagonals.
%C A185787 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:
%C A185787   1....2.....4.....7...11...16...22...29...
%C A185787   3....5.....8....12...17...23...30...38...
%C A185787   6....9....13....18...24...31...39...48...
%C A185787   10...14...19....25...32...40...49...59...
%C A185787   15...20...26....33...41...50...60...71...
%C A185787   21...27...34....42...51...61...72...84...
%C A185787   28...35...43....52...62...73...85...98...
%C A185787 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.
%C A185787 Examples:
%C A185787 row 1:  A000124
%C A185787 row 2:  A022856
%C A185787 row 3:  A016028
%C A185787 row 4:  A145018
%C A185787 row 5:  A077169
%C A185787 col 1:  A000217
%C A185787 col 2:  A000096
%C A185787 col 3:  A034856
%C A185787 col 4:  A055998
%C A185787 col 5:  A046691
%C A185787 col 6:  A052905
%C A185787 col 7:  A055999
%C A185787 diag. (1,5,...) ...... A001844
%C A185787 diag. (2,8,...) ...... A001105
%C A185787 diag. (4,12,...)...... A046092
%C A185787 diag. (7,17,...)...... A056220
%C A185787 diag. (11,23,...) .... A132209
%C A185787 diag. (16,30,...) .... A054000
%C A185787 diag. (22,38,...) .... A090288
%C A185787 diag. (3,9,...) ...... A058331
%C A185787 diag. (6,14,...) ..... A051890
%C A185787 diag. (10,20,...) .... A005893
%C A185787 diag. (15,27,...) .... A097080
%C A185787 diag. (21,35,...) .... A093328
%C A185787 antidiagonal sums:  (1,5,15,34,...)=A006003=partial sums of A002817.
%C A185787 Let S(n,k) denote the n-th partial sum of column k.  Then
%C A185787 S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
%C A185787 S(n,1)=n(n+1)(n+2)/6
%C A185787 S(n,2)=n(n+1)(n+5)/6
%C A185787 S(n,3)=n(n+2)(n+7)/6
%C A185787 S(n,4)=n(n^2+12n+29)/6
%C A185787 S(n,5)=n(n+5)(n+10)/6
%C A185787 S(n,6)=n(n+7)(n+11)/6
%C A185787 S(n,7)=n(n+10)(n+11)/6
%C A185787 Weight array of T: A144112
%C A185787 Accumulation array of T: A185506
%C A185787 Second rectangular sum array of T: A185507
%C A185787 Third rectangular sum array of T: A185508
%C A185787 Fourth rectangular sum array of T: A185509
%H A185787 Vincenzo Librandi, <a href="/A185787/b185787.txt">Table of n, a(n) for n = 1..1000</a>
%H A185787 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A185787 a(n)=n*(7*n^2-6*n+5)/6.
%F A185787 G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - _Vincenzo Librandi_, Jul 04 2012
%t A185787 f[n_,k_]:=n+(n+k-2)(n+k-1)/2;
%t A185787 s[k_]:=Sum[f[n,k],{n,1,k}];
%t A185787 Factor[s[k]]
%t A185787 Table[s[k],{k,1,70}]  (* A185787 *)
%t A185787 CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)
%o A185787 (Magma) [n*(7*n^2-6*n+5)/6: n in [1..50]]; // _Vincenzo Librandi_, Jul 04 2012
%Y A185787 Cf. A000027, A185788, A100182, A101165, A079824.
%K A185787 nonn,easy
%O A185787 1,2
%A A185787 _Clark Kimberling_, Feb 03 2011
%E A185787 Edited by _Clark Kimberling_, Feb 25 2023