A276587 -id:A276587 - cp's OEIS frontend

A276586 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), ...

1, 2, 3, 6, 8, 11, 30, 36, 44, 55, 210, 240, 276, 320, 375, 2310, 2520, 2760, 3036, 3356, 3731, 30030, 32340, 34860, 37620, 40656, 44012, 47743, 510510, 540540, 572880, 607740, 645360, 686016, 730028, 777771, 9699690, 10210200, 10750740, 11323620, 11931360, 12576720, 13262736, 13992764, 14770535

Offset: 0

Antti Karttunen, Sep 18 2016

			The top left corner of the array:
     1,     2,      6,       30,       210,       2310,        30030
     3,     8,     36,      240,      2520,      32340,       540540
    11,    44,    276,     2760,     34860,     572880,     10750740
    55,   320,   3036,    37620,    607740,   11323620,    253753500
   375,  3356,  40656,   645360,  11931360,  265077120,   7422334920
  3731, 44012, 686016, 12576720, 277008480, 7687412040, 235239464460

Transpose: A276587.
Topmost row: A002110, Leftmost column: A136104.
Cf. A007318, A276085.
Cf. also arrays A066117, A276588, A099884, A255483.

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 *)

P(n)=prod(i=1, n, prime(i));
T(n, k) = sum(j=0, n, binomial(n, j)*P(k + j));
for(n=0, 10, for(k=0, n, print1(T(k, n - k),", ");); print();) \\ Indranil Ghosh, Apr 11 2017

(define (A276586 n) (A276586bi (A002262 n) (A025581 n)))
(define (A276586bi row col) (A276085 (A066117bi (+ 1 row) (+ 1 col))))

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

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

A276589 Transpose of A276588.

Original entry on oeis.org

1, 3, 2, 11, 8, 6, 49, 38, 30, 24, 261, 212, 174, 144, 120, 1631, 1370, 1158, 984, 840, 720, 11743, 10112, 8742, 7584, 6600, 5760, 5040, 95901, 84158, 74046, 65304, 57720, 51120, 45360, 40320, 876809, 780908, 696750, 622704, 557400, 499680, 448560, 403200, 362880, 8877691, 8000882, 7219974, 6523224, 5900520, 5343120, 4843440, 4394880, 3991680, 3628800

Offset: 0

Antti Karttunen, Sep 19 2016

Rows give the successive first differences of A001339.

			The top left corner of the array:
     1,     3,     11,      49,      261,      1631,      11743
     2,     8,     38,     212,     1370,     10112,      84158
     6,    30,    174,    1158,     8742,     74046,     696750
    24,   144,    984,    7584,    65304,    622704,    6523224
   120,   840,   6600,   57720,   557400,   5900520,   68019240
   720,  5760,  51120,  499680,  5343120,  62118720,  780827760
  5040, 45360, 448560, 4843440, 56775600, 718709040, 9778048560

Index entries for sequences related to factorial base representation

Topmost row: A001339. For other rows and columns, see the information given in transpose A276588.

Cf. also A276587.

T[r_, c_]:=Sum[Binomial[r, k](1 + c + k)!, {k, 0, r}]; Table[T[r - c, c], {r, 0, 10}, {c, 0, r}] // Flatten (* Indranil Ghosh, Apr 11 2017 *)

T(r, c) = sum(k=0, r, binomial(r, k)*(1 + c + k)!);
for(r=0, 10, for(c=0, r, print1(T(r - c, c),", ");); print();) \\ Indranil Ghosh, Apr 11 2017

from sympy import binomial, factorial
def T(r, c): return sum([binomial(r, k) * factorial(1 + c + k) for k in range(r + 1)])
for r in range(11): print([T(r - c, c) for c in range(r + 1)]) # Indranil Ghosh, Apr 11 2017

(define (A276589 n) (A276588bi (A025581 n) (A002262 n))) ;; Code for A276588bi given in A276588.

cp's OEIS Frontend

A276586 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), ...

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