A001342 -id:A001342 - cp's OEIS frontend

A276588 Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*(1+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, 24, 30, 38, 49, 120, 144, 174, 212, 261, 720, 840, 984, 1158, 1370, 1631, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 362880, 403200, 448560, 499680, 557400, 622704, 696750, 780908, 876809, 3628800, 3991680, 4394880, 4843440, 5343120, 5900520, 6523224, 7219974, 8000882, 8877691

Offset: 0

Antti Karttunen, Sep 19 2016

			The top left corner of the array:
     1,     2,     6,     24,     120,      720,      5040,      40320
     3,     8,    30,    144,     840,     5760,     45360,     403200
    11,    38,   174,    984,    6600,    51120,    448560,    4394880
    49,   212,  1158,   7584,   57720,   499680,   4843440,   51932160
   261,  1370,  8742,  65304,  557400,  5343120,  56775600,  661933440
  1631, 10112, 74046, 622704, 5900520, 62118720, 718709040, 9059339520

Transpose: A276589.
Topmost row (row 0): A000142, Row 1: A001048 (without its initial 2), Row 2: A001344 (from a(1) = 11 onward), Row 3: A001345 (from a(1) = 49 onward), Row 4: A001346 (from a(1) = 261 onward), Row 5: A001347 (from a(1) = 1631 onward).
Leftmost column (column 0): A001339, Column 1: A001340, Columns 2-3: A001341 & A001342 (apparently).
Cf. A276075.
Cf. also arrays A066117, A276586, A099884, A255483.

T[r_, c_]:=Sum[Binomial[r, k](1 + c + k)!, {k, 0, r}]; Table[T[c, r - 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(c, r - 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(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 11 2017

(define (A276588 n) (A276588bi (A002262 n) (A025581 n)))
(define (A276588bi row col) (A276075 (A066117bi (+ 1 row) (+ 1 col)))) ;; Code for A066117bi given in A066117, and for A276075 under the respective entry.

A(row,col) = Sum_{k=0..row} binomial(row,k)*A000142(1+col+k).

A(row,col) = A276075(A066117(row+1,col+1)).

A076571 Binomial triangle based on factorials.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 24, 30, 38, 49, 65, 120, 144, 174, 212, 261, 326, 720, 840, 984, 1158, 1370, 1631, 1957, 5040, 5760, 6600, 7584, 8742, 10112, 11743, 13700, 40320, 45360, 51120, 57720, 65304, 74046, 84158, 95901, 109601

Offset: 0

Henry Bottomley, Oct 19 2002

			Rows start:
    1;
    1,   2;
    2,   3,   5;
    6,   8,  11,  16;
   24,  30,  38,  49,  65;
  120, 144, 174, 212, 261, 326;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
E. Biondi, L. Divieti, and G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35. See Table I.
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Columns include A000142, A001048, A001344, A001345, A001346, A001347.
Right hand columns include A000522, A001339, A001340, A001341, A001342.
Cf. A002627 (row sums), A099022.

A076571:= func< n,k| (&+[Binomial(k,j)*Factorial(n-j): j in [0..k]]) >;
[A076571(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 05 2023

A076571[n_, k_]:= n!*Hypergeometric1F1[-k,-n,1];
Table[A076571[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 05 2023 *)

def A076571(n,k): return sum(binomial(k,j)*factorial(n-j) for j in range(k+1))
flatten([[A076571(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 05 2023

T(n, k) = Sum_{j=0..k} binomial(k, j)*(n-j)!.
T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 0) = n!.
T(n, n) = A000522(n).
Sum_{k=0..n} T(n, k) = A002627(n+1).
From G. C. Greubel, Oct 05 2023: (Start)
T(n, k) = n! * Hypergeometric1F1([-k], [-n], 1).
T(2*n, n) = A099022(n). (End)

cp's OEIS Frontend

A276588 Square array A(row,col) = Sum_{k=0..row} binomial(row,k)*(1+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), ...

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula