A276586 - OEIS

%I #23 Apr 12 2017 23:25:10

%S 1,2,3,6,8,11,30,36,44,55,210,240,276,320,375,2310,2520,2760,3036,

%T 3356,3731,30030,32340,34860,37620,40656,44012,47743,510510,540540,

%U 572880,607740,645360,686016,730028,777771,9699690,10210200,10750740,11323620,11931360,12576720,13262736,13992764,14770535

%N Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*A002110(col+k), read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

%H Antti Karttunen, <a href="/A276586/b276586.txt">Table of n, a(n) for n = 0..989; the first 44 antidiagonals of array</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>

%F A(row,col) = Sum_{k=0..row} binomial(row,k)*A002110(col+k).

%F A(row,col) = A276085(A066117(row+1,col+1)).

%e The top left corner of the array:

%e      1,     2,      6,       30,       210,       2310,        30030

%e      3,     8,     36,      240,      2520,      32340,       540540

%e     11,    44,    276,     2760,     34860,     572880,     10750740

%e     55,   320,   3036,    37620,    607740,   11323620,    253753500

%e    375,  3356,  40656,   645360,  11931360,  265077120,   7422334920

%e   3731, 44012, 686016, 12576720, 277008480, 7687412040, 235239464460

%t primorial[n_] := Product[Prime[k], {k, 1, n}]; A[n_, k_] := Sum[Binomial[n, j]*primorial[k+j], {j, 0, n}]; Table[A[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 22 2017 *)

%o (Scheme)

%o (define (A276586 n) (A276586bi (A002262 n) (A025581 n)))

%o (define (A276586bi row col) (A276085 (A066117bi (+ 1 row) (+ 1 col))))

%o (PARI) P(n)=prod(i=1, n, prime(i));

%o T(n, k) = sum(j=0, n, binomial(n, j)*P(k + j));

%o for(n=0, 10, for(k=0, n, print1(T(k, n - k),", ");); print();) \\ _Indranil Ghosh_, Apr 11 2017

%Y Transpose: A276587.

%Y Topmost row: A002110, Leftmost column: A136104.

%Y Cf. A007318, A276085.

%Y Cf. also arrays A066117, A276588, A099884, A255483.

%K nonn,tabl

%O 0,2

%A _Antti Karttunen_, Sep 18 2016