%I #30 Feb 26 2023 08:18:54

%S 1,7,25,62,125,221,357,540,777,1075,1441,1882,2405,3017,3725,4536,

%T 5457,6495,7657,8950,10381,11957,13685,15572,17625,19851,22257,24850,

%U 27637,30625,33821,37232,40865,44727,48825,53166,57757,62605,67717,73100,78761,84707,90945,97482,104325,111481,118957,126760,134897,143375

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

%C 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 1....2.....4.....7...11...16...22...29...

%C 3....5.....8....12...17...23...30...38...

%C 6....9....13....18...24...31...39...48...

%C 10...14...19....25...32...40...49...59...

%C 15...20...26....33...41...50...60...71...

%C 21...27...34....42...51...61...72...84...

%C 28...35...43....52...62...73...85...98...

%C 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 Examples:

%C row 1: A000124

%C row 2: A022856

%C row 3: A016028

%C row 4: A145018

%C row 5: A077169

%C col 1: A000217

%C col 2: A000096

%C col 3: A034856

%C col 4: A055998

%C col 5: A046691

%C col 6: A052905

%C col 7: A055999

%C diag. (1,5,...) ...... A001844

%C diag. (2,8,...) ...... A001105

%C diag. (4,12,...)...... A046092

%C diag. (7,17,...)...... A056220

%C diag. (11,23,...) .... A132209

%C diag. (16,30,...) .... A054000

%C diag. (22,38,...) .... A090288

%C diag. (3,9,...) ...... A058331

%C diag. (6,14,...) ..... A051890

%C diag. (10,20,...) .... A005893

%C diag. (15,27,...) .... A097080

%C diag. (21,35,...) .... A093328

%C antidiagonal sums: (1,5,15,34,...)=A006003=partial sums of A002817.

%C Let S(n,k) denote the n-th partial sum of column k. Then

%C S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.

%C S(n,1)=n(n+1)(n+2)/6

%C S(n,2)=n(n+1)(n+5)/6

%C S(n,3)=n(n+2)(n+7)/6

%C S(n,4)=n(n^2+12n+29)/6

%C S(n,5)=n(n+5)(n+10)/6

%C S(n,6)=n(n+7)(n+11)/6

%C S(n,7)=n(n+10)(n+11)/6

%C Weight array of T: A144112

%C Accumulation array of T: A185506

%C Second rectangular sum array of T: A185507

%C Third rectangular sum array of T: A185508

%C Fourth rectangular sum array of T: A185509

%H Vincenzo Librandi, <a href="/A185787/b185787.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n)=n*(7*n^2-6*n+5)/6.

%F G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - _Vincenzo Librandi_, Jul 04 2012

%t f[n_,k_]:=n+(n+k-2)(n+k-1)/2;

%t s[k_]:=Sum[f[n,k],{n,1,k}];

%t Factor[s[k]]

%t Table[s[k],{k,1,70}] (* A185787 *)

%t CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 04 2012 *)

%o (Magma) [n*(7*n^2-6*n+5)/6: n in [1..50]]; // _Vincenzo Librandi_, Jul 04 2012

%Y Cf. A000027, A185788, A100182, A101165, A079824.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 03 2011

%E Edited by _Clark Kimberling_, Feb 25 2023